instability analysis of incompressibleopencavity flows · instability analysis of...

ESCUELA TECNICA SUPERIOR DE INGENIEROS AERONAUTICOS

UNIVERSIDAD POLITECNICA DE MADRID

Doctoral Thesis

Instability Analysis of Incompressible Open Cavity Flows

by

Fernando Meseguer Garrido

Aeronautical Engineer

Advisors

Eusebio Valero Sanchez and Javier de Vicente Buendıa

Madrid, October 2014

iii

Dedicado a mis padres

Abstract

The problem of the flow over an open cavity has been studied in depth in the literature,

both for being an interesting academical problem and due to the multitude of industrial

applications, like the landing gear of aircraft, or the water deposit of firefighter airplanes.

The different types of instabilities appearing in this flow studied in the literature are two:

the two-dimensional shear layer modes, and the three-dimensional modes that appear in

the main recirculating vortex inside the cavity.

In this thesis a parametric study in the incompressible limit of the problem is pre-

sented, using the linear stability analysis known as BiGlobal. This approximation allows

to obtain the global stability behaviour of the flow, and to capture both the morphological

features and the characteristics of the eigenmodes of the physical problem, whether they

are stable or unstable. The study presented here characterizes with great detail all the

relevant eigenmodes, as well as the hypersurface of instability on the parameter space of

the incompressible problem (Mach equal to zero, and variation of the Reynolds number,

the incoming boundary layer thickness, the length to depth aspect ratio of the cavity and

the spanwise length of the perturbation). The results allow to construct parametric re-

lations between the characteristics of the leading eigenmodes and the parameters of the

problem, like for example the one existing between the critical Reynolds number and its

characteristic length.

The numerical results presented here are compared with those of an experimental

campaign, with the main conclusion of said comparison being that the linear eigenmode are

present in the real saturated flow, albeit with some significant differences in the frequencies

of the experiments and those predicted by the theory. To try to determine the nature of

those differences a three-dimensional direct numerical simulation, analyzed with Dynamic

Mode Decomposition algorithm, was used to describe the process of saturation.

v

Resumen

El problema del flujo sobre una cavidad abierta ha sido estudiado en profundidad en la

literatura, tanto por el interes academico del problema como por sus aplicaciones practicas

en gran variedad de problemas ingenieriles, como puede ser el alojamiento del tren de

aterrizaje de aeronaves, o el deposito de agua de aviones contraincendios. Desde hace

muchos anos se estudian los distintos tipos de inestabilidades asociadas a este problema:

los modos bidimensionales en la capa de cortadura, y los modos tridimensionales en el

torbellino de recirculacion principal dentro de la cavidad.

En esta tesis se presenta un estudio parametrico completo del lımite incompresible

del problema, empleando la herramienta de estabilidad lineal conocida como BiGlobal.

Esta aproximacion permite contemplar la estabilidad global del flujo, y obtener tanto la

forma como las caracterısticas de los modos propios del problema fısico, sean estables o

inestables. El estudio realizado permite caracterizar con gran detalle todos los modos

relevantes, ası como la envolvente de estabilidad en el espacio parametrico del problema

incompresible (Mach nulo, variacion de Reynolds, espesor de capa lımite incidente, relacion

altura/profundidad de la cavidad, y longitud caracterıstica de la perturbacion en la di-

reccion transversal). A la luz de los resultados obtenidos se proponen una serie de rela-

ciones entre los parametros y caracterısticas de los modos principales, como por ejemplo

entre el Reynolds crıtico de un modo, y la longitud caracterıstica del mismo.

Los resultados numericos se contrastan con una campana experimental, siendo la prin-

cipal conclusion de dicha comparacion que los modos lineales estan presentes en el flujo

real saturado, pero que existen diferencias notables en frecuencia entre las predicciones

teoricas y los experimentos. Para intentar determinar la naturaleza de dichas diferencias

se realiza una simulacion numerica directa tridimensional, y se utiliza un algoritmo de

DMD (descomposicion dinamica de modos) para describir el proceso de saturacion.

vii

0 Acknowledgements ix

Acknowledgements

This work would have been impossible without the help and support of many people.

I want to thank all of them, for making this thesis happen.

First of all I want to thank my advisors, Eusebio Valero and Javier de Vicente for

their help and wisdom, and for this chance to start exploring the world of science. In this

regard I would also like to thank many other people for many fruitful discussions on a

great variety of topics that have led to this thesis, like professors Vassilis Theofilis, Leo

Gonzalez, Julio Soria or Jeremy Basley, among many others.

I also want to thank all my colleagues in the Applied Mathematics and Statistics De-

partment, for creating and incredible work environment of which I have the great pleasure

to be a part of. In the last few years I have had the luxury of entering in another wonderful

place, which is the Department of Aerospace Vehicles. I am immensely grateful for the

opportunity that I have been provided to develop a career in university, and the support

of my colleagues in the Design group has been outstanding.

Another aspect of which I am incredibly grateful for is the chance to travel as part

of this PhD, through the UPM funds for short stays, the help of the Applied Mathemat-

ics and Statistics Department or the Marie Curie Grant PIRSES-GA-2009-247651 “FP7-

PEOPLE-IRSES: ICOMASEF Instability and Control of Massively Separated Flows”. I

had the opportunity to meet great people around the globe, in Florida, Hawaii, Norway,

Italy or Australia. I have an extra word of acknowledgements for the many wonderful

friends I found in Australia, for making me feel at home while I could not be farther away

from home.

I also want to thank my friends and family, without whom this long journey would

have never been possible. The years have come and gone, but they have been always made

worthy by the people that surrounded me.

My last and greatest thanks is for my parents. I could not be more proud to be your

son.

0 Acknowledgements xi

Agradecimientos

Este trabajo habrıa sido imposible sin la ayuda de mucha gente. Quiero agradecerselo a

todos ellos, por hacer esta tesis posible.

En primer lugar quiero agradecer a mis tutores de tesis, Eusebio Valero y Javier de

Vicente por su ayuda y su sabidurıa, asi como por esta oportunidad de empezar a explorar

el mundo de la ciencia. En este aspecto tambien quiero agradecerle a mucha otra gente

las discusiones sobre temas muy variados que al final han conducido a esta tesis, como

los profesores Vassilis Theofilis, Leo Gonzalez, Julio soria o Jeremy Basley, entre otros

muchos.

Tambien quiero dar las gracias a todos mis companeros en el Departamento de Matematica

Aplicada y Estadıstica, por crear un entrono de trabajo increible del que tengo el inmenso

placer de formar parte. En los ulitmos anos he tenido la suerte de entrar en otro sitio

maravilloso, el Vepartamento de Aeronaves y Vehıculos Espaciales. Estoy inmensamente

agradecido por la oportunidad que se me ha brindado de desarrollar una carrera en la

universidad, y el apoyo de mis companeros en la catedra de Dibujo ha sido extraordinario.

Otro aspecto por el cual estoy increiblemente agradecido es por la oportunidad que he

tenido de viajar durante este doctorado, a traves de las ayudas del programa propio de

becas de la UPM, la asistencia del Departamento de Matematica Aplicada y estadıstica o

la beca Marie Curie PIRSES-GA-2009-247651 “FP7-PEOPLE-IRSES: ICOMASEF Insta-

bility and Control of Massively Separated Flows”. He tenido la oportunidad de conocer a

gente alrededor del mundo, en Florida, Hawaii, Noruega, Italia o Australia. Tengo que dar

las gracias especialmente a los numerosos amigos que conocı en Australia, por hacerme

sentir como si estuviera en casa cuando no podıa estar mas lejos de casa.

Quiero dar las gracias tambien a mis amigos y a mi familia, sin los cuales este largo

camino hubiera sido imposible. Los anos pasan uno tras otro, pero si merecen la pena es

gracias a la gente que me rodea.

Mi ultimo y mayor agradecimiento es para mis padres. No podrıa estar mas orgulloso

de ser vuestro hijo.

Contents

Abstract v

Resumen vii

Acknowledgements ix

Contents xiii

1 Introduction 1

2 Hydrodynamic Stability 11

2.1 Linear Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Non-Modal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Modal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 BiGlobal Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Numerical Methods 21

3.1 Short Review of Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Base Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 BiGlobal instability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Three-dimensional Direct Numerical Simulation . . . . . . . . . . . . . . . . 29

3.5 Dynamic Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 The three-dimensional dynamics 35

xiii

xiv CONTENTS

4.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 On complex conjugate eigenvalues . . . . . . . . . . . . . . . . . . . 39

4.2 Global instability validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Comparison with Bres & Colonius . . . . . . . . . . . . . . . . . . . 44

4.2.2 Comparison with 3D DNS . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Parametric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 Preliminary considerations on the envelope of stability . . . . . . . . 50

4.3.2 Spanwise wavenumber (β) . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.3 Reynolds number (ReD) . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.4 Incoming boundary-layer thickness (θ0/D) . . . . . . . . . . . . . . . 57

4.3.5 Aspect ratio (L/D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.6 Eigenmode morphology . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.7 Synthesis of the results . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 The two-dimensional limit 77

5.1 Convergence of BiGlobal analysis on β → 0 . . . . . . . . . . . . . . . . . . 77

5.2 Shear layer modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.1 Characteristic frequency . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 On eigenmode amplification . . . . . . . . . . . . . . . . . . . . . . . 81

6 Experimental campaign 87

6.1 The experimental campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2.1 Validity of eigenmodes in the saturated regime . . . . . . . . . . . . 89

6.2.2 Effects of three-dimensional boundary conditions . . . . . . . . . . . 93

6.2.3 On the symmetry breakings . . . . . . . . . . . . . . . . . . . . . . . 95

CONTENTS xv

6.3 Concluding remarks on the experimental campaign . . . . . . . . . . . . . . 95

7 Preliminary study on saturation 99

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.2 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.1 Regime I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.2 Regime II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.3 Regime III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.3.4 Regime IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.3.5 Regime V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 Concluding remarks on the study on saturation . . . . . . . . . . . . . . . . 109

8 Summary and Future Directions 111

xvi CONTENTS

List of Figures

2.1 Transient growth due to non-orthogonal combination of two vectors decay-

ing in time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Schematic description of the two-dimensional open cavity and problem pa-

rameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Variation with the Reynolds number of the incoming boundary layer thick-

ness at the leading edge of the cavity, θ0/D, for constant boundary layer

thickness at the start of the computational domain θ−1/D = 0.0337. . . . . 37

4.3 Schematic description of the 3-D open cavity and problem parameters. . . 38

4.4 Spanwise velocity component, w, isosurfaces for a stationary eigenmode. . 40

4.5 Spanwise velocity component isosurfaces for a travelling eigenmode for dif-

ferent combinations of α1 and α2. On top, pulsating perturbation. The

others are structures that are right-travelling (middle row) or left-travelling

(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Effect of small variation of the flow conditions on the stability. Ampli-

fication vs β, with full symbols corresponding to variations in incoming

boundary layer thickness and empty symbols to variations in ReD. . . . . . 43

4.7 Comparison between Bres and Colonius [20] results (black diamonds) and

present BiGlobal analysis solution (Mode I in red circles, Mode II (bifur-

cated) in blue circles) for the nominal conditions of case A. Amplification

rate as a function of the spanwise length on the left and of the dimensionless

frequency on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 Amplification versus β for Case A. Mode I in red circles, Mode II (bifur-

cated) in blue circles and Bres and Colonius [20] results in black rhombi. . . 46

xvii

xviii LIST OF FIGURES

4.9 Temporal evolution of the maximum value of the spanwise velocity pertur-

bation obtained by DNS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.10 3D Visualization of spanwise velocity at ReD = 1500 and β = 6: Leading

disturbance obtained using BiGlobal analysis on the left; DNS solution on

the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.11 3D Visualization of spanwise velocity disturbance obtained at ReD = 2400

using BiGlobal analysis (left) and DNS at t = 400 (right). . . . . . . . . . 49

4.12 Left:variation of the eigenvalues amplification and frequency with β. L/D =

2, Re= 2300, θ−1 = 0.0337, β = 4.4 in red, β = 4.8 in green, β = 5.2 in blue

and β = 5.6 in black. Right: Variation of the curves σ versus β with the

Reynolds number. All cases are for L/D = 2 and θ−1 = 0.0337, Re= 2300

in red, Re= 2400 in green, Re= 2500 in blue and Re= 2600 in black. . . . . 52

4.13 Amplification, σ, versus β and ReD of the three leading modes with positive

amplification in the open cavity flow of aspect ratio L/D = 2 with θ−1 =

0.0337. Mode I in red, Mode II in blue and the third mode in white. . . . . 53

4.14 Neutral stability curves of the three leading modes in the open cavity flow

of aspect ratio L/D = 2 with θ−1 = 0.0337. Mode I in red, Mode II in blue

and the third mode in white. Critical values are cited in table 4.6. . . . . . 54

4.15 Dependence on Reynolds number of the amplification rate, σ, of the leading

eigenmodes in the open cavity flow of aspect ratio L/D = 2 with θ−1 =

0.0337. Mode I in red, Mode II in blue and the third mode in white. . . . 55

4.16 Neutral stability curves of the two leading modes in the open cavity flow of

aspect ratio L/D = 2 with θ−1 = 0.0337 in circles, approximate maxima in

rhombus. Mode I in red, Mode II in blue. . . . . . . . . . . . . . . . . . . . 56

4.17 Neutral curves in ReD vs θ0/D for the critical β. Mode I in red symbols,

Mode II (bifurcated) in blue symbols. L/D = 2 in circles and L/D = 3 in

squares. Highlighted point corresponds with the nose of the most unstable

mode in figure 4.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.18 Neutral curves for θ−1/D = 0.0337 of Mode I. Aspect ratio varying from

L/D = 1 (higher ReD numbers) to L/D = 3 (lower ReD numbers) with the

values detailed in Table 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

LIST OF FIGURES xix

4.19 Neutral curves for θ−1/D = 0.0337 of Mode II. Aspect ratio varying from

L/D = 1 (higher ReD numbers) to L/D = 3 (lower ReD numbers) with the

values detailed in Table 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.20 Neutral curves for θ−1 = 0.0337. From left to right, and from top to bottom,

L/D = 1.0, L/D = 1.1, L/D = 1.2, L/D = 1.3, L/D = 1.4, L/D = 1.5,

L/D = 2.0 and L/D = 3.0. Mode I in red, Mode II (bifurcated) in blue,

third mode in white and fourth mode in black. . . . . . . . . . . . . . . . . 61

4.21 Neutral curves in the ReD vs L/D for the critical β and θ−1 = 0.0337. First

mode in red, second mode in blue, with circles being low β and rhombi high

β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.22 From left to right and top to bottom, w velocity component isosurfaces of

Mode I for, L/D = 1.1 (ReD = 3167, β = 12.45), L/D = 1.1 (ReD = 3630,

β = 6.56), L/D = 1.5 (ReD = 1472, β = 5.96), L/D = 2 (ReD = 1150,

β = 5.62) and L/D = 3 (ReD = 865, β = 5.30) respectively. In all cases

the spanwise length shown is Lz = 2π/β = 2D. . . . . . . . . . . . . . . . . 64

4.23 From left to right and top to bottom, w velocity component isosurfaces for

the stationary branch of Mode II for, L/D = 1 (ReD = 3421, β = 13.07),

L/D = 1.3 (ReD = 2069, β = 10.95), L/D = 1.5 (ReD = 1708, β = 10.11),

L/D = 2 (ReD = 1471, β = 9.86) and L/D = 3 (ReD = 1342, β = 10.06)

respectively. In all cases the spanwise length shown is Lz = 2π/β = 2D. . . 65


the travelling branch of Mode II for, L/D = 1.1 (ReD = 3884, β = 6.63),

L/D = 1.3 (ReD = 2608, β = 5.63), L/D = 1.5 (ReD = 1853, β = 4.84),

L/D = 2 (ReD = 1523, β = 4.45) and L/D = 3 (ReD = 1638, β = 4.61)



the third mode, L/D = 1.2 (ReD = 3476, β = 11.61), L/D = 2 (ReD =

2207, β = 10.34) and the fourth mode of L/D = 3 (ReD = 1442, β = 2.92)


4.26 From left to right and top to bottom, u velocity component isosurfaces for

Mode I. Time with respect of the final time t/tf = [0; 0.25; 0.5; 0.625; 0.75; 0.875; 0.94; 1]

In all cases the spanwise length shown is Lz = 2π/β = 2D. . . . . . . . . . 68

xx LIST OF FIGURES

4.27 From left to right and top to bottom, v velocity component isosurfaces for

Mode I. Time with respect of the final time t/tf = [0; 0.25; 0.5; 0.625; 0.75; 0.875; 0.94; 1]

In all cases the spanwise length shown is Lz = 2π/β = 2D. . . . . . . . . . 69


Mode I. Time with respect of the final time t/tf = [0.75; 0.875; 0.94; 1] In

all cases the spanwise length shown is Lz = 2π/β = 2D. . . . . . . . . . . . 70

4.29 Neutral curves in ReL vs θ0/D for the critical β. Mode I in red symbols,

Mode II (bifurcated) in blue symbols. L/D = 2 in circles and L/D = 3 in

squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.30 Qualitative representation of the spanwise velocity component of the low β

lobe of Mode I for L/D = 1.2, 1.5, 2 and 3, from top to bottom and left to

right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.31 Dependence with L/D and LP /D of the critical ReD for the critical β (top)

and said β of maximum amplification (bottom). All data with θ−1/D =

0.0337. Mode I in red symbols, Mode II in blue symbols, with circles for

low β and rhombi high β. In lines, the predicted curves of equation 4.4,

and the hyperbolic law described in § 4.3.7, and in small empty symbols, of

same shape and color, the equivalent LP /D of the same points. . . . . . . . 73

4.32 Strouhal number for the main oscillating modes versus the length-to-depth

aspect ratio of the cavity, both in logarithmic scale. Mode I, Mode II and

the third unstable mode are in red circles, blue circles and empty circles

respectively. The other two, in triangles, are stable modes for the range of

parameters studied. In black rhombi the dominant frequency of Bres and

Colonius [20], and the range of frequencies obtained by Basley et al. [16] on

the crossed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Evolution of the growth rate of the two least stable eigenvalues on the two

dimensional limit, in a range of β ∼ 10−2 to β ∼ 10−9 for L/D = 2,

ReD = 1500 and the constant δ−1. . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Velocity profiles of the reconstructed 2D flow with the first shear layer mode.

Streamwise velocity on the top, and wall normal velocity on the bottom. . . 79

LIST OF FIGURES xxi

5.3 Variation of the dimensionless frequency (Strouhal based on cavity length

StL) of the two least stable eigenmodes (circles ◦ and rhombus � respec-

tively) in the 2D limit with the incoming boundary-layer thickness. Results

for L/D = 2 in empty symbols (◦), and L/D = 3 in full symbols(•). In

grey shades, the range of Strouhal numbers obtained by Sarohia [90], and

in red symbols single points of several L/D = 2 works. The 2M2 run of

Rowley et al. [87] in squares (�), the Bres [19] M= 0.3 run as a rhombus

( ), the lower runs of Yamouni et al. [112] in triangles (�). Points from

Basley et al. [16] as circles (©). . . . . . . . . . . . . . . . . . . . . . . . . . 80


StL) of the two least stable eigenmodes (circles ◦ and rhombus � respec-

tively) in the 2D limit with the length of the cavity dimensionalized with the

incoming boundary-layer thickness, L/θ0. Results for L/D = 2 in empty

symbols (◦), and L/D = 3 in full symbols(•). In in red symbols single

points of several works: the 2M2 run of Rowley et al. [87] in squares (�),

the Bres [19] M= 0.3 run as a rhombus ( ), the lower runs of Yamouni

et al. [112] in triangles (�) and points from Basley et al. [16] as circles (©). 81

5.5 Streamwise velocity profiles of the first shear layer mode, for ReD = 2400

and θ−1/D = 0.0337. Different length domains, from top to bottom xout =

8, 13, 19 and 21 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 82


StL) and amplification σ of the two least stable eigenmodes with the change

of domain length xout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.7 Neutral curves in ReD vs θ0 for the two least stable two-dimensional modes,

in circles ◦ and rhombus � respectively. Three-dimensional Mode I and

Mode II in red and blue, as in figure 4.17. Domain length kept constant at

xout = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.1 Sketch of the experimental set-up. Dimensions are given in millimetres. The

laser sheet (at y = −0.1D) is represented in a close-up on the L = 2D –

shaped cavity. High resolution images require three cameras to span the

cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xxii LIST OF FIGURES

6.2 Main flow features at ReD = 2400 for both experimental and linear stability analysis.

(Top) BiGlobal unstable eigenvalues (dots) and qualitative schematic depiction of the

most energetic modes in the experiments (shades) in the β-St plane. (Bottom) Velocity

fields related to the four representative modes highlighted in the upper figure. Left

column corresponds to global Fourier modes from the experimental dataset. Right column

presents the reconstructed flow using BiGlobal analysis. For each mode, streamwise

velocity (top) and spanwise velocity (bottom) are shown. . . . . . . . . . . . . . . . 91

6.3 Streamwise profiles of streamwise velocity U/U0 for case A (ReD = 1500)

on the left and case B (ReD = 2400) on the right. The profile obtained

of the 2D base-flow used by BiGlobal analysis is extracted from the range

−0.12 � y/D � −0.09 (black), to represent the uncertainty on the position

and thickness of the laser-sheet. The profile issued of the 3D mean-flow,

experimentally measured in the zx-plane at y = −0.1D is extracted from

the range −3 � z/D � 3 (blue), to take into account spanwise variations. . 94

7.1 Neutral curves for the L/D = 2 cavity in the ReD vs β plane, and selected

Case B (top). StD vs β map of unstable eigenmodes for Case B, and

selected β by the periodicity conditions of the DNS computations (bottom). 101

7.2 Temporal evolution of the absolute value of the span-wise velocity compo-

nent in the control point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3 DMD modes on regime II. On the top, situation in the StD vs β plane

(left) of DMD modes A and B (right). On the bottom, BiGlobal mode

corresponding with point A (Mode II for β = 12). . . . . . . . . . . . . . . 104

7.4 Two instantaneous flowfields in region III (top). Composition of the two

linear modes that yields a similar flowfield (bottom). . . . . . . . . . . . . 106

7.5 DMD modes on regime III. Situation in the StD vs β plane of DMD modes

A and C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.6 DMD modes on regime IV. Situation in the StD vs β plane of DMD and

modes A, C, E and D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.7 DMD modes on regime V. Situation in the StD vs β plane of DMD and

modes A, C, E and F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

List of Tables

4.1 Parameters characterising the inflow Blasius boundary layer and associated

location of the primary, (xC , yC), and secondary, (xc, yc), vortices. . . . . . 37

4.2 Parameters for Case A (ReD = 1500). B&C stands for the nominal condi-

tions of Bres & Colonius 08. ED for the Experimental Data of Chapter 6,

with associated uncertainties. LSA for the values of Linear Stability Anal-

ysis. The values of δ∗0 and θ0 are calculated with an equivalent rig to the

one employed in Chapter 6 for B&C and LSA. . . . . . . . . . . . . . . . . 43

4.3 Parameters of the nominal conditions for Case B (ReD = 2400). . . . . . . 44

4.4 Amplification rate (σ) and frequency (StD) of the leading BiGlobal mode

and the linear part of the DNS, for both cases. . . . . . . . . . . . . . . . . 50

4.5 Range of the parameters studied . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Critical parameters of the first three modes for the open cavity flow with

aspect ratio L/D = 2 and θ−1 = 0.0337. . . . . . . . . . . . . . . . . . . . . 52

4.7 Critical parameters of the first three modes for each of the aspect ratios

and for constant δ−1/D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 The parameters C1 and C2 of equation (4.4) for the different modes(left)

and the parameters K1 and K2 of equation (4.5) for the different modes

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xxiii

1Introduction

Flow over open cavities is of theoretical and practical interest from both a hydrodynamic

and an aeroacoustic point of view. Two-dimensional cavities, i.e. those in which flow in the

lateral spatial direction is considered to be homogeneous, may be encountered in planar

or axisymmetric form, embedded in incompressible or compressible flow over semi-infinite

(open) or confined (closed) domains. Geometric details of a two-dimensional open cavity

configuration, such as its cross-sectional shape, length-to-depth aspect ratio and the form

of either or both cavity lips, as well as the relative dimension of the cavity compared

with a characteristic length scale of the oncoming flow, make description of open cavity

flow non-unique. A finite spanwise extent of the cavity, i.e. three-dimensionality of the

cavity geometry itself, adds yet another dimension to this multi-parametric problem, as

does turbulence, the latter being essential for the description of most industrially relevant

flows.

From a theoretical point of view, progress in the understanding of the complex open

cavity flow dynamics is made by addressing the relatively simple two-dimensional rectangu-

lar cavity configuration at moderate Reynolds numbers. In this context, a steady laminar

two-dimensional boundary-layer on a flat surface encountering an open cavity separates

as a free shear-layer from the upstream cavity corner. Two independent Reynolds num-

bers characterise this flow, one based on the incoming boundary-layer flow properties and

a second based on the cavity dimensions. The dependence on the incoming boundary

layer thickness can be studied as a length parameter, keeping the effect of the velocity

on the problem only on the Reynolds number based on cavity depth. At the relatively

low Reynolds number values of interest in the present thesis framework, at which two-

dimensional flow remains laminar, and depending on the cavity aspect ratio, the free

shear-layer either impinges upon the downstream cavity corner in a steady or unsteady

manner, or curves toward the cavity floor to form a closed recirculation bubble. In ei-

ther case, a new boundary layer forms on the downstream cavity wall, starting from the

downstream cavity corner, which may itself be steady or unsteady.

1

2

Flow instabilities in an open cavity have been investigated in the past from the point

of view of understanding the basic physical mechanisms involved, but also motivated

by the multitude of industrial applications in which this configuration arises, such as

open roofs on automobiles, landing gear in aircraft or weapon bays. Theoretical and

numerical expediency have limited most analyses of open cavity flows to strictly two spatial

dimensions, or three-dimensional domains in which spanwise periodicity is assumed. Fully

non-periodic three-dimensional analysis still has a prohibitive computational cost.

The majority of early theoretical work on the open cavity has revolved around the

two-dimensional flow/acoustic resonance producing self sustained oscillations in the shear

layer (Powell [79], Rockwell [81], Rockwell and Naudascher [84]). In compressible flow,

small amplitude oscillations in the shear layer impinging at the downstream edge of the

cavity generate pressure disturbances that travel upstream and reinforce the shear layer

oscillations. This feedback mechanism results in vortex-shedding at the leading edge, which

locks onto a characteristic frequency that follows the semi-empirical formula described by

Rossiter [86], established for compressible flows.

Subsequent works postulated that in the incompressible regime this feedback can be

considered as instantaneous, and in the experimental work of Sarohia [90] two peaks of

frequency that match the prediction of the Rossiter mechanism were found. Later Gharib

and Roshko [40] observed in their work on axisymmetric cavities that, as the incoming

boundary-layer thickness decreased in relation to the depth of the cavity, there was a

substantial change in behaviour of the cavity oscillations, characterised by a large scale

shedding from the cavity leading edge; the authors termed this finding the wake mode.

Similar transitions have been observed by Pereira and Sousa [76] in two-dimensional in-

compressible Direct Numerical Simulations (DNS) with laminar incoming boundary layer.

Sipp and Lebedev [97] studied the bifurcation of the shear layer modes in the incompress-

ible confined cavity of aspect ratio L/D = 1 on supercritical flow conditions, using weakly

non-linear analysis. Following that work, Barbagallo et al. [10] and Sipp [96] delved into

the closed and open-loop control problem, respectively, at those same conditions.

Still in the incompressible regime, the self-sustained oscillations follow the primarily

two-dimensional geometry of the shear layer. In that regime, Colonius et al. [25] showed the

first two-dimensional amplitude function in a two-dimensional open cavity, demonstrating

a Tollmien−Schlichting wave emanating on the downstream wall and being connected with

instability inside the open cavity (see also Supplemental Appendix 4 of Theofilis [102]).

The work of Rowley et al. [87] provided further insight regarding the onset of shear-layer

oscillations from a steady flow and their nonlinear interactions, developing a criterion to

1 Introduction 3

predict the onset of disturbances from the steady flow. Later, Theofilis and Colonius

[103, 104] revisited the open cavity and, applying the residuals algorithm (Gomez et al.

[42], Gomez et al. [43], Theofilis [100]) recovered the same Tollmien−Schlichting eigenmode

as well as an acoustic mode connecting hydrodynamic perturbations inside the cavity

with pressure fluctuations having their origin at the downstream cavity corner. Recently,

Yamouni et al. [112] have given a detailed description of the effect of compressibility on the

interaction of the acoustic feedback and resonance using global instability analysis in a two-

dimensional cavity at a particular set of flow conditions and one Reynolds number. The

authors confirmed that the shear layer modes correspond to the beginning of branches of

those global modes whose frequencies evolve with Mach number as described by Rossiter’s

semi-empirical formula, as is supported by previous findings by Sarohia [90].

The thorough study of the two-dimensional self sustained oscillations led to the dis-

covery of three dimensional structures in the flow, with frequencies far smaller than the

ones of the Rossiter modes. Several studies reported these lower frequencies, mainly as a

modulation of the two-dimensional shear layer modes (Koseff and Street [62], Neary and

Stephanoff [71], Rockwell and Knisely [82]). The hypothesis that these modes were the

result of non-linear interaction between Rossiter modes was rebutted by Cattafesta III

et al. [22] and Kegerise et al. [57].

The aforementioned work by Theofilis and Colonius [104] also contained some of the

first linear computations of three dimensional instabilities inside the cavity. The authors

used a two dimensional base flow multidomain computation, and applying the appropriate

boundary conditions to the upper side on the cavity subdomain they showed a structure

consisting of a perturbation that coils around the main recirculating vortex. That mon-

odomain approach in the eigenvalue problem solution requires a numerical simplification

on the boundary condition, and has the problem of being unable to reproduce the interac-

tions of the cavity flow with the shear layer and the external flow. The first non-conforming

spectral multidomain approach for the numerical solution for the two-dimensional BiGlobal

eigenvalue problem, in this case for the lid driven cavity was presented in de Vicente et al.

[28], paving the road for future multidomain approaches in the open cavity.

The first full three-dimensional global instability analysis of compressible flow over

a rectangular open cavity was performed by Bres and Colonius [20]. In said work the

authors establish that the low frequency mode corresponds to three-dimensional struc-

tures associated to the main recirculation vortex inside the cavity, and they found that

these three-dimensional perturbations were dominant for some flow conditions, and that

they are independent from the shear layer modes. These three-dimensional structures are

4

sometimes known in the literature as centrifugal modes, and may qualitatively be related

to instability modes seen in the lid-driven cavity (Albensoeder et al. [4], Gonzalez et al.

[46], Theofilis [99]). Bres and Colonius [20] also confirmed the previously observed modu-

lation of the two-dimensional shear layer modes by the lower frequencies of the centrifugal

modes. Further discussion on the nature of the interaction between the two types of modes

can be found in Basley et al. [17]

It is interesting to note that such spanwise dynamics arise in the recirculating flow

in shear-driven as well as lid-driven cavities (Albensoeder et al. [5], de Vicente et al.

[29], Gonzalez et al. [46]), that is, regardless of the shear-layer oscillations.

There also exist in the literature numerous works on flow visualisation and charac-

terisation of these three-dimensional open cavity instabilities. Recently, Faure et al. [36]

aimed to understand the three-dimensional flow morphology, and characterise experimen-

tally the three-dimensional structures in the open cavity, confirming that the dynamical

structures are independent of shear layer instabilities. Basley et al. [15] used time-resolved

particle image velocimetry (PIV) to extract the spatial distribution of the most charac-

teristic frequencies in the incompressible open cavity with two different aspect ratios, and

also identified once again the presence of the aforementioned shear layer modes in the

incompressible regime. Alizard et al. [6] focused on the global instability analysis of open

flows using a domain decomposition matrix-free method. One of the benchmark problems

the latter authors employed is the square cavity, where, at a single Reynolds number, the

three-dimensional spanwise periodic leading perturbations were identified.

Despite these previous works a complete study on three-dimensional incompressible

cavity flow is still missing. Recently, the work of de Vicente et al. [31] presented a combined

theoretical, numerical and experimental work on global instability of the open cavity in

incompressible regime. It also includes validation and verification of the instability analysis

results, the focus being on the three-dimensional structures, dominant in the studied flow

regime. This work is partially presented here, as well as the totality of Meseguer-Garrido

et al. [66]. The aim of those publications is to present an in-depth extension of the

theoretical results presented in de Vicente [27], revisiting incompressible flow over the

spanwise periodic rectangular cavity. A systematic global (BiGlobal, using the terminology

of Theofilis et al. [105]) instability analysis was performed, taking into account the effect

of parameters that, no doubt on account of the cost of this type of analysis, have either

been studied incompletely or been altogether ignored in the literature. The incoming

boundary-layer thickness, the Reynolds number based on cavity depth and the spanwise

wavenumber are varied independently, for a range of length-to-depth aspect ratios, until

1 Introduction 5

the most interesting part of the spectrum containing all amplified global eigenmodes is

revealed. Also, these studies aim to bring more insights into the early stages of the three-

dimensional dynamics associated with centrifugal effects around the main recirculation

inside the cavity, in a flow regime in which the shear-layer modes are not dominant.

The unstable three-dimensional perturbations are identified and characterised from both

numerical and experimental approaches.

Moreover, the fruitful combination of linear analysis and experiments leads to a deeper

knowledge of the characteristics and evolution of these modes from their onset to their

final presence in the nonlinear saturated state. In order to delve more deeply on that

evolution from the linear regime to the saturated state, three-dimensional direct numerical

simulations were performed, and analyzed using a dynamic mode decomposition technique,

as presented in Vinha et al. [110].

Data-sequences of snapshots collected from numerical simulations or experiments can

be used to approximate the inherent fluid flow into dynamic modes, allowing then the

identification of the relevant coherent structures in the flow. The most commonly used

data-based techniques so far are the Fourier Transform analysis, the Proper Orthogonal

Decomposition (POD) and, more recently, the Dynamic Mode Decomposition (DMD).

The first approach is particularly efficient when dealing with periodic sampled data-fields.

However, it loses accuracy when dealing with more complex and time-dependent fluid

flows. With the use of POD one can identify the relevant structures in the flow ranked

by their energy content but, since the different POD modes are orthogonal in space, their

temporal behavior is characterized by the presence of multiple frequencies. For a detailed

discussion about the use of Fourier Transform analysis, POD, and alternative data-based

decomposition techniques on the identification of coherent structure in the fluid flow, the

reader is directed to Mezic [67] and Bagheri [8].

The DMD allows the extraction of spatial modal structures from a flow field, where

each identified dynamic mode is associated to a single and unique frequency, consequence of

the orthogonalization in time of the decomposition. The present technique is based on the

Koopman analysis of a dynamical system by Rowley et al. [88], aiming to approximate the

Koopman modes and eigenvalues of a linear infinite dimensional operator that describes

that system, even if its dynamic behavior is nonlinear. In this case, DMD retrieves the

structures of a linear tangent approximation to the underlying fluid flow (Schmid [92]).

Nonetheless, for a system behaving linearly, the extracted DMD modes are expected to

match the global stability modes. Moreover Rowley et al. [88] analytically demonstrated

that performing a DMD over periodic solutions leads to the same decomposition obtained

6

via a discrete temporal Fourier Transform. Contrary to the POD, the DMD does not rank

the extracted coherent structures in terms of energy content. However, their amplitudes

provide a feedback about the individual contribution of a specific mode to the original

data-set (Schmid [93]), providing the DMD the possibility of obtaining models of lower

complexity (Jovanovic et al. [55]), as it already happens with the POD.

The DMD is a recent decomposition technique that is becoming more and more attrac-

tive in post-processing of numerical and experimental results, mainly due to its easiness of

implementation, efficient data analysis, inherent low computational cost, and possibility

of application to large data-sets or to sub-domains of a flow region. Some implementations

of this tool for cavity problems can be found in the literature, for example in Gomez et al.

[42] or in Ferrer et al. [37]. This method has also demonstrated superior performance over

other traditional data-based decomposition techniques for oscillatory dominated problems

(Schmid [93]) and for fluid flows presenting strong peaks in the spectrum (Mezic [67]).

Nonetheless, DMD still has some relevant limitations, as recognized in Bagheri [8]. Ac-

cording to this reference, there is yet no validation between Koopman and DMD modes

for chaotic and noisy high-Reynolds number flow-data and, based on the work of Duke

et al. [33], the Dynamic Decomposition can be sensitive to the presence of noise in the

data-field. Furthermore, Muld et al. [70] observed no particular differences between the

POD and the DMD modes in a flow characterized by a broad frequency spectrum and no

dominant spectral peaks. Finally, the DMD may not guarantee the best approximation of

the flow field, opening new ways for improved variants of the original algorithm based on

optimization techniques, like the ones proposed by Jovanovic et al. [55] and Chen et al.

[23].

Regarding the nature of the BiGlobal analysis, from a numerical point of view solutions

have been obtained in the last decades using a variety of approaches. From the point of

view of solution methodology, straightforward serial (see Theofilis [101] for a review) or

parallel (Rodrıguez and Theofilis [85]) implementations of a subspace iteration variant, can

be considered, or employment of the widely used (Bagheri et al. [9], Bres and Colonius

[20], Crouch et al. [26], Fietier and Deville [38], Janke and Balakumar [54], Wintergerste

and Kleiser [111]) implementation of the Implicitly Restarted Arnoldi Method (IRAM)

in the ARPACK (Lehoucq et al. [64]) software package for the solution of large scale

eigenvalue problems. On the other hand, from the point of view of matrix formation,

existing methodologies for the solution of the BiGlobal eigenvalue problem can be classified

in two categories, one in which the matrix is formed, stored and processed using dense

(serial or parallel) linear algebra technology, and another known as matrix-free/time-

1 Introduction 7

stepper algorithms (Abdessemed et al. [2], Bagheri et al. [9], Barkley et al. [13], Sherwin

and Blackburn [95]).

Regarding spatial discretization methods of the BiGlobal eigenvalue problem, early ap-

plications studied simple two-dimensional domains in which the numerical discretization

techniques employed were straightforward extensions of those used in the solution of classic

one-dimensional linear stability eigenvalue problems, using spectral methods (Pierrehum-

bert [77], Tatsumi and Yoshimura [98], Zebib [113]). At the same time, finite-element

methods were also used for the solution of the BiGlobal eigenvalue problem by Jackson

[52] and Morzynksi and Thiele [69]. Although the low order of those methods limits the

accuracy of the solution, they are not restricted to the single-domain two-dimensional

grids employed in the early spectral analyses. This lack of accuracy can be critical in the

case or problems with sharp gradients, as is the case of the BiGlobal eigenmodes at high

Reynolds numbers, leading to the use of unstructured meshes of ever-increasing density in

order to achieve convergence (Gonzalez et al. [44]), trading off the efficiency of a high-order

method in favor of the flexibility offered by the unstructured mesh discretization.

The use of high-order accurate, flexible and efficient numerical methods in order to

solve the BiGlobal EVP is needed. Such an approach has been introduced in the seminal

work of three-dimensional instability in the wake of a circular cylinder by Barkley and Hen-

derson [12], Henderson and Barkley [49] in the form of spectral-element discretization on

structured meshes. The first application of a spectral/hp−element method (Karniadakis

and Sherwin [56]) to the study of a global instability problem on unstructured meshes was

that of Theofilis et al. [106], who recovered instability in the wake of a NACA0012 airfoil

as the leading BiGlobal eigenmode of the steady wake flow. While the aforementioned

spectral/hp−element analyses utilized time-stepping concepts (Tuckerman and Barkley

[109]), Gonzalez et al. [45] have discussed matrix formation and storage as an alternative

technique for the solution of the same problem in the context of spectral/hp−element

discretization. The method here used was explained in depth in de Vicente [27], as the

codes and numerical tools developed there are the basis for all the stability computations

presented here.

8

The organization of the present thesis is as follows:

Chapter 2 contains a brief explanation of the linear instability theory, focused on

the temporal stability problems here analyzed. The nature of the BiGlobal instability

analysis is explained, resulting on the analysis of spanwise periodic three-dimensional

global disturbances that evolve over a two-dimensional steady base flow.

Chapter 3 is a compendium of the numerical methods employed. Section 3.2 explains

the two-dimensional spectral element method used to integrate the laminar Navier-Stokes

equations in order to obtain the base flows, Section 3.3 details the BiGlobal eigenvalue

problem solver, Section 3.4 the three-dimensional dicontinuous galerking spectral element

code, and Section 3.5 the dynamic mode decomposition tool.

Chapter 4 describes the problem at hand and puts in context previous findings. First,

Section 4.2 contains the extensive validation of the tool with the literature, and with

the three-dimensional DNS code. Results of parametric analyses are shown in Section 4.3,

where the critical Reynolds numbers and the leading perturbations at all conditions studied

are identified. A synthesis of three-dimensional global instability analysis results obtained

herein provides analytical expressions for the relation between critical Reynolds number

and both incoming boundary-layer thickness and cavity aspect ratio, which are given in

Section 4.3.7. The connection between the characteristic frequency of the leading global

eigenmodes and the cavity aspect ratio is revealed from these results.

Chapter 5 contains the results regarding the two-dimensional instabilities. Although

the shear layer modes are not the focus of this thesis, some interesting results are shown,

as well as the concordance of the results with those present in the literature.

Chapter 6 explains the main results of the experimental campaign presented in de

Vicente et al. [31]. Results from both linear stability analysis and experiments enable the

onset of centrifugal instabilities inside the cavity to be related to the final saturated state

of the flow. To that end, Section 6.2 discusses the relationship between global Fourier

modes from the experiments with respect to global eigenmodes. Section 6.3 summarizes

the most significant conclusions obtained.

Chapter 7 is a preliminary study on the saturation phenomena, using the three-

dimensional DNS computations, and comparing the DMD analysis of the snapshots ob-

tained with the linear analysis results.

1 Introduction 9

Most of the material in the present thesis, as well as additional specific details have

been presented in the following list of publications:

• Journal Articles

◦ de Vicente, J. & Basley, J. & Meseguer-Garrido, F. & Soria, J. & Theofilis,

V. “Three-dimensional instabilities over a rectangular open cavity: from linear

stability analysis to experimentation”, J. Fluid Mech. (2014), vol. 748, pp.

189-220.

◦ Meseguer-Garrido, F. & de Vicente, J. & Valero, E. & Theofilis, V. “On lin-

ear instability mechanisms in incompressible open cavity flow”, J. Fluid Mech.

(2014), vol. 752, pp. 219-236.

◦ Vinha, N. & Meseguer-Garrido, F. & de Vicente, J. & Valero, E. “A Dynamic

Mode Decomposition of the saturation process in the open cavity flow”, sub-

mitted to Special Issue of Aerospace Science and Technology on ”Massively

Separated Flows”.

• Conference Papers

◦ Gonzalez L. & Theofilis, V. & Meseguer-Garrido, F. “Applications of High

Order Methods to Vortex Instability Calculations”, International Conference on

Spectral and High Order Methods (ICOSAHOM), 22-26 June 2009, Trondheim,

Norway.

◦ Meseguer-Garrido, F. & de Vicente, J. & Valero, E. & Theofilis, V. “Effect of

Aspect Ratio on the Three-Dimensional Global Instability Analysis of Incom-

pressible Open Cavity Flows”, 6th AIAA Theoretical Fluid Mechanics Confer-

ence, 26-30 June 2011, Honolulu, Hawaii.

◦ Meseguer-Garrido, F. & de Vicente, J. & Valero, E. “Three-dimensional analysis

of incompressible flow over an open cavity using direct numerical simulation:

from linear to saturated regime”,Instability and Control of Massively Separated

Flows, 5980, June 5-8 2013, Prato, Italy.

2Hydrodynamic Stability

Contents

2.1 Linear Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Non-Modal Instability . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Modal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2.1 Local Instability . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2.2 Non-Local Instability . . . . . . . . . . . . . . . . . . . 17

2.2 BiGlobal Linear Theory . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Linear Instability

Linear stability theory concerns the evolution of small disturbances superposed on a

steady- or time dependent basic state. A solution, called “base flow”, is said to be stable

if an infinitely small variation will alter only by an infinitely small quantity the basic flow

at some future time.

||U(x, 0)− U∗(x, 0)|| < δ → ||U(x, t)− U∗(x, t)|| < ε (2.1)

Otherwise the solution is said to be unstable. The solution is said asymptotically stable

if it converges to the original solution when it is perturbed:

||U(x, t)− U∗(x, t)|| → 0 ,as, t → ∞ (2.2)

Linear stability analysis in the framework of the solutions of the incompressible Navier-

Stokes equations is related with the decomposition of any flow quantity q in a basic steady

flow q plus a disturbance component q.

q = q+ εq (2.3)

where the flow state is defined by the velocity components in each direction and pressure

q(x, t) = (u(x, t), P (x, t)), with u(x, t) = (u(x, t), v(x, t), w(x, t)) and x = (x, y, z) is the

11

12 2.1 Linear Instability

vector comprising the spatial coordinates. The steady and perturbed states are defined the

same way. Introducing this decomposition into the dimensionless incompressible Navier-

Stokes equations:

∂u

∂t+ (u · ∇)u =

1

Re∇2u−∇P (2.4)

∇ · u = 0

yields,

∂u

∂t+

∂u

∂t+(u ·∇)u+(u ·∇)u+(u ·∇)u+(u ·∇)u =

1

Re∇2u+

1

Re∇2u−∇P −∇P (2.5)

∇ · u+∇ · u = 0

Stability concerns the evolution of small disturbances. The initial decomposition (2.3)

has been fed into the Navier-Stokes equations. O(1) terms, corresponding to the basic

flow, have been canceled while O(ε2) term, the non-linear (u · ∇)u, is neglected. The

perturbed problem is stated only in O(ε) terms.

This gives the linearized problem:

∂u

∂t+ (u · ∇)u+ (u · ∇)u =

1

Re∇2u−∇P (2.6)

∇ · u = 0

From this point, two different analyses derive. Non-modal analysis derives from re-

writing (2.6) as:∂q

∂t= LNSE (Re, q) q (2.7)

where LNSE (Re, q) is the linear operator depending on the Reynolds number and the

base flow. Stability is governed by the properties of the operator. On the other hand

modal analysis, based on classical linear theory, establishes that if the basic flow is steady,

the coefficients of the linearized problem are independent of t, therefore by separation of

variables, the general solution is a linear superposition of normal modes, each of the form:

q(x, t) = q(x)exp(λt) (2.8)


2.1.1 Non-Modal Instability

The non-modal approach takes the so called state space equation (2.6) as a starting point.

The linearized Navier-Stokes equations can be written as,

dq

dt= LNSE (Re, q) q (2.9)

where LNSE is the former linear operator. This operator is non-normal (it does not

commute with its adjoint) and consequently it is not necessarily diagonalizable. This

implies that the solution, total sum of modes, can experience an initial non-linear growth

(transient growth), although every individual mode decay with time.

This growth in the early stages could provoke that the energy of the disturbance

increases, eventually reaching a critical value leading to transition. The theoretically

predicted modal decay may be overruled depending on the initial perturbation. Linked

to this is the concept of optimal disturbances, i.e., which initial state cause the maximum

energy growth within a time period (Butler and Farrell [21], Farrell and Ioannou [34],

Schmid and Henningson [91], Trefethen et al. [108]).

This linear behavior can be explained from a numerical point of view. Once the linear

operator has been spatially discretized (in this work using spectral collocation) the system

(2.9) behaves like a linear system with constant coefficients in time where solution can be

expressed as

q = exp(A(Re, q)t)q0 (2.10)

where A is the m-dimensional matrix discretizing the spatial differential operator in the

m-points of the computational domain. Stability of the solution (2.10) only depends on

the fundamental matrix Φ(t) = exp(A(Re, q)t), particularly on the eigenvalues of matrix

A(Re, q).

In the general case of n different eigenvalues λi each of them with algebraic multiplicity

ki, associated to each eigenvalue it is possible to find a set of linearly independent solutions

of the kind

Qij = exp(λit)

j−1∑l=0

tl

l!vj−l j = 1, . . . , ki. (2.11)

where vj is a sequence of generalized eigenvectors for the eigenvalue λi.

The set Qij i = 1, . . . n conforms a system of fundamental solutions associated to the

basis of generalized eigenfunctions. These fundamental solutions are the columns of the

basis of the fundamental matrix Φ(t). So each solution of (2.10) can be expressed like


Figure 2.1: Transient growth due to non-orthogonal combination of two vectors decaying in time.

q =

n∑i=1

ki∑j=1

Qijq0(l) l = 1, . . . ,m. (2.12)

The most significant implication comes from the fact that every solution is formed

by two parts: one exponential and other polynomial. The exponential factor is clearly

dominant when time becomes large, however initially when t is small enough, polynomial

behavior could dominate, and, depending on the initial solution q0, perturbations could

grow in a nonlinear way.

Other source of initial linear growth comes from the non-orthogonality of the eigenfunc-

tions of (2.9) even when LNSE is diagonalizable and all the eigenvalues have negative real

part. Solutions of (2.10) can be expressed as a linear combination of the eigenmodes:

q =

n∑i=1

q0(i)exp(λit) (2.13)

Although the eigenmodes are characterized by decreasing length, due to the nature of

their associated eigenvalues, the solution vector may grow in the early stages as seen in

2.1 taken from Schmid and Henningson [91].

2.1.2 Modal Instability

This work is focused in the modal study of instability of open cavity flows. As mentioned

this analysis is based on the decomposition of the flow quantities in two parts, (i) the

steady or time periodic basic state upon which (ii) small disturbances are permitted to

develop.


In the particular case of time periodic basic state a similar assumption can be made

using Floquet theory, which seeks the eigenvalues of a time periodic operator. These eigen-

values, also known as Floquet multipliers, determine the development of small-amplitude

perturbations during one period of evolution. In this work only steady states have been

considered. To deal with time periodic basic states in the instability analysis of incom-

pressible flows Barkley and Henderson [12] or more recently Karniadakis and Sherwin [56]

and Abdessemed [1] works could shed light over this problem.

The eigenvalues and their corresponding eigenmodes can be found by substituting (2.8)

into (2.6). That leads to a generalized eigenvalue problem (EVP) of the form

Aq = λBq (2.14)

Numerical discretization of the spatial directions and imposition of the appropriate

boundary conditions complete the matrix formulation. Using this approach, the stability

of the basic flow is determined by the eigenvalues of (2.14). If Re(λ) < 0 for all the

eigenvalues the basic flow is asymptotically stable and unstable if Re(λ) > 0 for at least

one eigenvalue.

Different stability analysis derive from the dimensional decomposition of the fluid vari-

ables in the two mentioned states. In it most general form separability of time and spatial

directions in the governing equations leads to the following decomposition of fluid vari-

ables:

q(x, t) = q(x) + εq(x, t) (2.15)

where q = (u, P )T = (u, v, w, P )T , q = (u, P )T = (u, v, w, P )T and x=(x,y,z). Once

the decomposition (2.15) has been substituted into the governing equations one may write

q(x, t) = q(x)exp(−i ωt) (2.16)

This stability analysis, where physical space is three-dimensional and the time-periodic

disturbances are inhomogeneous in all the three directions, is named TriGlobal linear

instability analysis (Theofilis [101]). However the three-dimensional eigenvalue problem

resulting from this decomposition is extremely expensive, and presently only within the

possibilities of the most advanced supercomputing facilities at relevant Reynolds numbers.


A wide range of simplifications to the flow instability problem are possible in order

to make it solvable. From the simplest parallel-flow assumption to the BiGlobal insta-

bility analysis, main tool of this work, complexity increases for both the basic flow and

disturbances.

2.1.2.1 Local Instability

The greatest level of simplification comes from considering homogeneous basic flows in

the x and z directions, i.e. neglecting the dependence of the basic flow q on x and

z. Consequently velocity component v is also neglected. These assumptions, known as

parallel-flow assumptions, permit the decomposition

q(x, y, z, t) = q(y) + εq(y)exp{i (αx+ βz − ωt)} (2.17)

where α and β are the wavenumbers in the spatial x and z directions, and ω stands for

the frequency in the temporal instability analysis framework. Gaster’s relation permits

the transformation between the temporal and spatial problems (Gaster [39]).

The substitution into the incompressible flow equations leads to a system of equations

for the amplitude functions of the disturbance velocity components and the disturbance

pressure. Pressure can be eliminated resulting an equation for the normal velocity. To

complete the flow field description another equation is included for the normal vorticity.[1

Re

(d2

dy2− (

α2 + β2))2

− iαu

(d2

dy2− (

α2 + β2))

+ iαd2u

dy2

]v =

(d2

dy2− α2

)ωv

(2.18)[iαu− 1

Re

(d2

dy2− (

α2 + β2))2

]η + iβ

du

dyv = iωη (2.19)

with the boundary conditions v = dvdy = η = 0 at solid walls and in the free stream, where

η =∂u

∂z− ∂w

∂x(2.20)

is the normal vorticity.

Equation (2.18) for the normal velocity is the well known Orr-Sommerfed equation

(Drazin and Reid [32]). In the adopted formulation the equation constitutes an one-

dimensional eigenvalue problem to recover the frequencies, eigenvalues of the temporal sta-

bility problem, and the associated eigenmodes. Meanwhile (2.19), is the Squire equation,

an eigenvalue problem for normal vorticity fed with the solutions of the Orr-Sommerfeld

equation.


2.1.2.2 Non-Local Instability

The Orr-Sommerfeld equation provides the exact eigenmodes for some flows like pressure

driven Poiseuille channel flow or the shear driven Couette flow. Nevertheless, it is now

known that the predictions of Orr-Sommerfed equation in a variety of flows, like Poiseuille

and Couette flows are wrong; for physically relevant results to be obtained in such prob-

lems, the non-modal approach discussed earlier must be adopted.

Quasi-parallel flows like Blasius boundary layers requires the modeling of the distur-

bance quantities being spatially inhomogeneous to recover proper solutions.

To deal with non-parallel boundary layers flows, an extension of the parallel theory

is the Parabolized Stability Equation (PSE) (Bertolotti et al. [18], Herbert [50]). In this

analysis, analogously to the parallel classic linear theory, one spatial direction is resolved.

By contrast, the basic flow is permitted to grow in one or both remaining spatial directions.

The flow decomposition yields

q(x, y, z, t) = q(y) + εq(y)exp(i (

∫ x

x0

α(ξ)dξ + βz − ωt)) (2.21)

When the ansatz (2.21) is replaced in the Navier Stokes and continuity equation a

system of equations for the disturbance is set. A normalization condition is needed for

closure in order to ensure that the streamwise variation of the amplitude function q remains

small. Further information on this kind of instability analysis can be found in Herbert

[51].

2.2 BiGlobal Linear Theory

Between the simplified parallel-flow assumption and the extremely costly TriGlobal insta-

bility analysis, BiGlobal instability analysis represents a intermediate and affordable step

in the instability analysis. The main difference with the parallel flow assumption is that,

here, three-dimensional space comprises an inhomogeneous two-dimensional domain which

is extended periodically in z characterized by a wavelength Lz. The BiGlobal framework

involves the substitution of a decomposition of any of the independent flow variables, e.g.

the three velocity components, temperature and pressure q(x, y, z, t) = (u, v, w, θ, P )T into

the (incompressible or compressible) equations of motion. All quantities are considered

to be composed of an O(1) steady two-dimensional basic state and small-amplitude O(ε)

unsteady three-dimensional perturbations

18 2.2 BiGlobal Linear Theory

q(x, y, z, t) = q(x, y) + εq(x, y, z, t) (2.22)

The separability of temporal and spatial derivatives in (2.6) permits introduction of an

explicit temporal dependence of the disturbance quantities into these equations. Spatially,

disturbances q(x, y, z, t) are considered three dimensional but only two directions are in-

homogeneous (x, y), while in the third spanwise direction z, disturbances are permitted

to assume a harmonic expansion. These assumptions lead to the ansatz:

q(x, y, z, t) = q(x, y)ei (βz−ωt) (2.23)

for the determination of the complex eigenvalue:

ω = 2πStDU∞

D+ iσ. (2.24)

In the temporal framework used, the Strouhal number (St) represents a dimension-

less frequency and σ is the amplification/damping rate of the disturbance sought, while

barred and hatted quantities denote basic and disturbance flow quantities, respectively.

The two-dimensional domain is extended periodically in the homogeneous z-direction with

the periodicity length Lz = 2π/β associated with the real wavenumber β. The complex

eigenvalues and the corresponding eigenmodes q(x, y), are recovered when the decompo-

sition 2.23 is substituted in the linearized equations 2.6:

[L − (Dxu)] u− (Dyu) v −DxP = −i ωu, (2.25)

− (Dxv) u+ [L − (Dyv)] v −DyP = −i ωv, (2.26)

− (Dxw) u− (Dyw) v + Lw − i βP = −i ωw, (2.27)

Dxu+Dyv + i βw = 0. (2.28)

where Dx = ∂/∂x, Dy = ∂/∂y and

L =1

Re

(∂2

∂x2+

∂2

∂y2− β2

)− u

∂

∂x− v

∂

∂y− iβw. (2.29)

An important simplification comes from the absence of the z-component in the velocity

profile of the basic flow. Homogeneity in the spanwise direction (β real) and redefinitions:

i w → w i ω → ω (2.30)


result in the following generalized real nonsymmetric partial derivative eigenvalue prob-

lem:

[L∈ − (Dxu)] u− (Dyu) v −DxP = ωu, (2.31)

− (Dxv) u+ [L∈ − (Dyv)] v −DyP = ωv, (2.32)

L∈w − βP = ωw, (2.33)

Dxu+Dyv − βw = 0. (2.34)

where

L∈ =1

ReD

(∂2

∂x2+

∂2

∂y2− β2

)− u

∂

∂x− v

∂

∂y. (2.35)

So the EVP (2.14) can be re-written in the BiGlobal instability analysis framework as

Aq = ωBq (2.36)

There are some implications due to this formulation. The first one, mainly numerical,

is the reduction in the storage requirements of the real EVP compared with the original

complex problem. Halving the storage required for the spatial discretization results in

the ability to address flow instability at higher resolutions, what is essential in case of

higher Reynolds numbers or strong gradients in the flow. The second one is merely an

interpretative issue. Real part of the eigenvalue ω corresponds to the growth rate, σ, while

the imaginary part is related to the oscillation frequency. If the growth rate is positive the

amplitude of the eigenmode will grow exponentially in time and the basic flow is unstable.

The problem is completed with the imposition of boundary conditions for the distur-

bance quantities. At solid walls no-slip boundary conditions are imposed for the velocity

components while homogeneous Neumann condition has been found to be the best for

the disturbance pressure. Boundary conditions for the disturbance pressure do not exist

physically, but compatibility is ensured with the choice of normal first derivative equal

to zero. The imposition of appropriate boundary conditions for open boundaries is not a

straightforward problem, especially when flow leaves the computational domain. In Chap-

ter 3, devoted to the numerical methods the election of appropriate boundary conditions

will be discussed.

3Numerical Methods

Contents

3.1 Short Review of Spectral Methods . . . . . . . . . . . . . . . . . 21

3.2 Base Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 BiGlobal instability analysis . . . . . . . . . . . . . . . . . . . . 28

3.4 Three-dimensional Direct Numerical Simulation . . . . . . . . . 29

3.5 Dynamic Mode Decomposition . . . . . . . . . . . . . . . . . . . 32

The main objective in this Chapter is to detail the different numerical tools employed

to obtain the results presented in subsequent chapters. In order to do that a tool is needed

to obtain approximated numerical solutions for the instability analysis of the Navier Stokes

equations in a cavity. To perform this analysis an eigenvalue problem (EVP) has to be

solved. The EVP requires a prior steady solution of the two-dimensional Navier Stokes

equations (Base flow DNS). To find accurate solutions of these problems, the spatial

integration is performed via a spectral Chebyshev multi-domain collocation method. In

order to validate and compare the results of the BiGlobal analysis a three-dimensional

direct numerical simulation (3D DNS) tool was used. Also, to process the data provided

by the three-dimensional DNS a Dynamic Mode Decomposition (DMD) tool was used.

3.1 Short Review of Spectral Methods

Spectral methods are members of the broader category of weighted Residual Methods.

The main distinguishing feature of these methods is the selection of mutually orthogonal

global functions as basis for the series expansion of the solution of the differential equation.

Spectral methods have been extensively used in computational fluid dynamics due to

their accuracy and efficiency in the simulation of fluid flows. These features come from

the mathematical properties of orthogonal basis in a Hilbert space. In said space, an

orthogonal basis is a numerable set of vectors, mutually orthogonal, that are complete in

this space. In the particular case of the Hilbert space L2w([a, b]), set of square integrable

21

22 3.1 Short Review of Spectral Methods

functions in [a, b], completeness of the basis means that any function f(x) ∈ L2w([a, b]) can

be approximated as closed as desired using an expansion in terms of that basis.

f(x) =

∞∑k=1

akϕk(x), (3.1)

where the coefficients ak are obtained projecting f(x) over the elements in the basis. The

projection is defined using an inner product,

(u(x), v(x))w =

∫ b

au(x)v(x)w(x)dx, (3.2)

with u(x), v(x) ∈ L2w([a, b]) and w(x) a weight function. The choice of the weight function

and the characteristics of the mathematical or physical problems determine the orthogonal

basis used.

The most common family of orthogonal polynomials used in spectral methods is the

set of trigonometric functions,

ϕk(x) = eikx, (3.3)

also called Fourier basis. The expansion in Fourier series is the most usual for the ap-

proximation of periodic functions. The fast decay of the k-th coefficient, (faster than any

inverse power of k for smooth function with all its derivatives periodic) implies that the

Fourier series represents the best approximation of the function. This feature is referred as

“spectral accuracy” of the Fourier method. However, the characteristic oscillatory behav-

ior of the Fourier expansion of a function in the neighborhood of a point of discontinuity

(Gibbs phenomenon), makes this approximation non-profitable for non-periodic problems.

In order to avoid this restriction in the boundary the use of some other expansion series

is required.

To obtain other families of orthogonal basis functions, two processes can be used:

First one is the Gram-Schmidt orthogonalization procedure. When this process is

applied to the polynomial system {1, x, x2, x3, ...}, depending on the domain [a, b] over

which the problem is defined and the weight w(x) used to characterize the inner product

(3.2) in L2w([a, b]), one can find:

• Jacobi polynomials. Set of orthogonal polynomials in [−1, 1]. These family corre-

sponds to the choice w(x) = (1− x)α(1 + x)β

J(α,β)k (x) =

1

2k

k∑i=0

(k + α

i

)(k + β

k − i

)(x− 1)i(x+ 1)k−i (3.4)


In this broad class of orthogonal polynomials are included the Legendre polynomials,

when α = β = 0, and the Chebyshev polynomials corresponding to α = β = −1/2.

• Laguerre polynomials. Family of orthogonal polynomials in [0,∞), with the weight

function w(x) = xαe−x

• Hermite polynomials. The domain of definition for this set of functions is (−∞,∞),

while the weights are w(x) = 1

Gram-Schmidt procedure results in a set of orthogonal functions that could be used

as the orthogonal basis for the expansion of any function in L2w([a, b]).

The other approach to systems of orthogonal polynomials comes from fact that each

of the aforementioned families of orthogonal polynomials in [−1, 1] are solution of the

second-order linear differential Sturm-Liouville problem.

{−(pφ′)′ + qφ = λwφ in the interval (1,−1),

plus boundary conditions for φ(3.5)

where

p(x) ∈ C1[−1, 1]; q(x), w(x) ∈ C[−1, 1]

p(x) > 0, w(x) > 0 ∀x ∈ (−1, 1)

The Sturm-Liouville problems consist on determining the values of λ, given p(x) and q(x)

continuous functions and weight w(x), for which non trivial solutions φ(x) exist.

The eigenfunctions of a Sturm-Liouville eigenvalue problem are of special interest in

spectral methods due to some of their properties.

• The eigenvalues of a regular Sturm-Liouville problem satisfy

λ1 < λ2 < · · · < λi < · · · (3.6)

• There is a unique eigenfunction φi(x) corresponding to each eigenvalue λi

• The eigenfunctions are mutually orthogonal using the inner product:

(φi(x), φj(x))w = 0, if i �= j (3.7)

• The eigenfunction φi(x) has exactly i− 1 zeros in (−1, 1)

24 3.1 Short Review of Spectral Methods

These properties of the eigenfunctions of a Sturm-Liouville problem are the reason why

this problem is so important in the spectral methods theory. First and second properties

imply that each eigenvalue has algebraic multiplicity one. This guarantees the linear in-

dependence of the eigenfunctions. Moreover the set of mutually orthogonal eigenfunctions

constitutes a family of orthogonal polynomials and the zeros of these polynomials are

preferred points in the collocation method. Nevertheless the most crucial feature of the

eigenfunctions of the Sturm-Liouville problem is the order in the sequence of eigenvalues

(property one) and its relation with the asymptotic decay of the expansion coefficients of

a function in terms of the orthogonal eigenfunctions. This property is the base under the

so-called spectral accuracy, the small quantity of terms in the expansion needed to achieve

a good approximation.

To link the Sturm-Liouville problem with the Chebyshev polynomials, it is necessary

to make a distinction between regular and singular Sturm-Liouville problems.

When p(x) vanishes at the boundaries the Sturm-Liouville problem is singular. The

only polynomials which can be solution of a singular Sturm-Liouville problem are included

in the general class of Jacobi polynomials.

In the particular case when p(x) = (1 − x2)(1/2), q(x) = 0 and w(x) = (1 − x2)(−1/2),

the singular Sturm-Liouville problem becomes the Chebyshev equation,

(x2 − 1)d2u

dx2+ x

du

dx= λ2u (3.8)

The Chebyshev polynomials of first kind are the eigenfunctions of this problem. (The

Chebyshev polynomials of second kind derive from choosing w(x) = w−1(x))

In this work only Chebyshev polynomials have been used in the study of the instability

of flows in cavities. There are several reasons for this choice, these polynomials maintain

the spectral accuracy property, are suitable for non-periodic problems in finite domains

and their roots and peaks play an important role in the optimal approximation theory. In

what follows, only this kind of polynomials will be considered.


3.2 Base Flow Calculations

BiGlobal instability analysis requires a steady two dimensional basic state in which time

disturbances are superimposed. This base solution is computed by the integration of the

laminar Navier-Stokes equations. Consistency with the methodology chosen for the insta-

bility analysis impelled to use a spectral method in the calculation of the two-dimensional

base flow.

The Navier-Stokes equations governing the motion of a viscous incompressible fluid

are set up using velocity-pressure formulation. Numerical methods utilized for spatial-

and time-discretization are presented and discussed. The Chebyshev collocation method is

applied for the spatial discretization using the multi–domain technique to handle nontrivial

geometries. The time integration technique, a semi-implicit coupled scheme where artificial

compressibility is added in the continuity equation, will be detailed.

Finally initial and boundary conditions required for the different problems solved are

also discussed. The problematic open boundary, where the fluid enters or leave the com-

putational domain, will be treated carefully.

The steady two-dimensional basic flow required for the BiGlobal instability analysis

of the open cavity flow is obtained using a semi-implicit artificial compressibility method

detailed in de Vicente et al. [29].

Using this method, the Navier-Stokes equations governing the motion of a viscous

incompressible fluid become:

∂u

∂t+ (u · ∇)u =

1

ReD∇2u−∇P, (3.9)

ε∂P

∂t+∇ · u = 0, (3.10)

where u represents the velocity vector comprising the two velocity components in the

streamwise and wall normal directions (u, v), P is the pressure and ε is an arbitrarily small

parameter for the artificial compressibility component. Chebyshev spectral collocation

technique has been chosen for the spatial discretization, while time advance has been

performed using a semi-implicit Euler scheme leading to the following system of discretized

equations:

26 3.2 Base Flow Calculations

un+1 − un

dt− 1

ReD∇2un+1 + Pn+1

x = −(ununx + vnuny ),

vn+1 − vn

dt− 1

ReD∇2vn+1 + Pn+1

y = −(unvnx + vnvny ),

εPn+1 − Pn

dt− un+1

x − vn+1y = 0. (3.11)

This semi-implicit numerical scheme treats convection terms in the Navier-Stokes equa-

tion explicitly but pressure and viscous terms implicitly. This approach avoids solving a

nonlinear problem at each time step while preserving the stability properties of implicit

solvers. The choice of an Euler method for time advancing is due to its simple implemen-

tation and its good stability properties when a steady solution is sought.

The main drawback of this kind of method is the loss of accuracy before the steady

state is reached. This is not an issue here, since in the BiGlobal analysis methodology a

steady base solution is sought.

The spectral multi-domain methodology implemented is here fully exploited by de-

composing the computational domain into rectangular sub-domains. The presence of a

laminar boundary layer upstream of the cavity, the shear layer developed from the leading

edge, as well as the boundary layer on the downstream cavity wall all require a fine grid

to be properly resolved. A stretching law permits the concentration of the Chebyshev

collocation points in the most interesting part of the domain and consequently leads to a

better resolution.

The computational domain must be defined so that it is large enough to isolate as

much as possible the numerical effects of imposing boundary conditions for open flows.

In the open cavity flow four different kinds of boundaries appear: wall, inflow, far-field

and outflow. First issue concerns the implementation of appropriate boundary conditions.

For viscous wall-bounded flows no-slip boundary conditions are adequate for velocity, while

homogeneous Neumann conditions for the pressure derivative in the normal direction to

the wall are applied.

The velocity components of the laminar boundary layer at the inflow of the computa-

tional domain are calculated by integrating the Blasius equation. Several Blasius profiles

have been computed and used in order to obtain a range of basic flows at different Reynolds

numbers and with different incoming boundary layer thickness, as required by the subse-

quent instability analysis. In the part of the study in which the boundary layer thickness

was not modified it was fixed to a reference value at the start of the computational domain


(δ−1). This reference value for the incoming Blasius profile was chosen to be the same as

in Rowley et al. [87] and Bres and Colonius [20].

This condition, originally chosen to match the works of the aforementioned authors,

is unconventional, in the sense that the parameter kept constant is not the one that

is supposed to dominate the behaviour of the problem (which would be the boundary

layer thickness at the leading edge of the cavity), but a computational one (the incoming

boundary layer thickness at the start of the computational domain). That means that the

constant incoming boundary layer thickness results correspond in reality not to a plane

of such constant thickness, but to a manifold in the parameter space, as it was described

before. The general effect on the results of this is, nevertheless, small, and in any case the

effect of said incoming boundary layer thickness has also been thoroughly studied, as will

be seen in what follows.

So, the following boundary condition is defined at the inflow boundary:

u = uBlasius, (3.12)

v = vBlasius, (3.13)

while, as in standard boundary layer flows, Dirichlet conditions for velocity field lead

to homogeneous Neumann conditions for pressure in the normal direction. The far-field

boundary denotes the upper artificial limit, far enough from the boundary layer, where the

streamwise component of the velocity vector is equal to a characteristic velocity. Normal

velocity is considered constant in the y−direction.

The choice and implementation of the boundary condition in the downstream limit

of the computational domain has been the most critical issue in the convergence of the

steady basic flow calculation. Several works dealing with this open boundary numerical

treatment have been published since 70’s (Gresho and Sani [47], Kobayashi et al. [58],

Orszag and Israeli [72], Orszag et al. [73]). The boundary condition which has shown the

best performance consists of forcing the second derivatives of the flux velocity components

to vanish in the outflow while keeping the pressure constant with a Dirichlet boundary

condition (Gresho and Sani [47], Sani and Gresho [89]):

∂2u

∂x2= 0,

∂2v

∂x2= 0,

P = P0. (3.14)

Regarding the boundary conditions utilised, on the solid walls the viscous conditions

discussed by (de Vicente et al. [29]) have been applied. At the inflow boundary a Blasius

28 3.3 BiGlobal instability analysis

profile is imposed, corresponding with the appropriate Reynolds number and incoming

boundary layer thickness, while the outflow, xout, and far-field, yfar, boundaries have

been placed at large distances from the cavity in order to minimise numerical effects due

to the imposition of artificial boundary conditions for open flows (de Vicente et al. [30]).

Typical parameters chosen are yfar = 5 and xout = 8.

3.3 BiGlobal instability analysis

As it was explained in section 2.2 linear stability analysis in the BiGlobal framework

involves the substitution of a decomposition of any of the independent flow variables, e.g.

the three velocity components, temperature and pressure q(x, y, z, t) = (u, v, w, θ, P )T into

the (incompressible or compressible) equations of motion. The obtention of the complex

eigenvalue from the generalised real non-symmetric eigenvalue problem in schematic form,

Aq = ωBq. (3.15)

Solutions of (3.15) are sought subject to the following boundary conditions for the

disturbance quantities. At solid walls the no-slip boundary condition u = v = w =

0 is imposed on the velocity components, alongside a compatibility condition for the

disturbance pressure, where an homogeneous Neumann condition is the most adequate.

As it was previously mentioned, boundary conditions for the disturbance pressure do not

exist physically, but compatibility is ensured with the choice of normal first derivative

equal to zero.

∂P

∂n= 0. (3.16)

Inflow boundary is treated in the same way, considering there are no velocity fluctua-

tions across this boundary; homogeneous Neumann boundary conditions are imposed there

for the pressure perturbations. At the outflow boundary Neumann boundary conditions

are applied on all the perturbation variables,

∂u

∂n=

∂v

∂n=

∂w

∂n=

∂P

∂n= 0. (3.17)

Far-field, the top boundary of the computational domain, has been initially treated like

the inflow condition. This choice has been proved to be correct for the leading cavity

modes, whose structure is confined within the cavity. On the other hand, these conditions


strongly affect the growth rate of the two dimensional shear layer modes, as it will be

explained in section 5.

For a detailed discussion on the boundary conditions used in this analysis the inter-

ested reader is referred to the work by de Vicente [27]. Spatial domain is decomposed

using the same multi-domain methodology explained in the base flow section, each do-

main discretized using Chebyshev collocation technique. The resulting two-dimensional

BiGlobal eigenvalue problem is solved by Krylov subspace iteration in two stages. First,

the matrix discretizing the EVP is LU-decomposed (though never formed explicitly); sec-

ond, the decomposition is fed into an Arnoldi iteration with shift and invert methodology

to recover the leading eigenvalues closest to the stability bound.

3.4 Three-dimensional Direct Numerical Simulation

The details of the three-dimensional direct numerical simulation used to validate the

BiGlobal results are as follows.

The compressible laminar Navier-Stokes equations constitute a system of partial dif-

ferential equations which can be shortly written in vector form as:

∂U

∂t+∇ · F(U) = 0, (3.18)

where U represents the vector of conservative variables and F(U) represent the 3D fluxes;

including convective and diffusive in the three coordinate directions.

The computationally-demanding nature of the Navier-Stokes solution, in the stability

analysis context, leads to the selection of high-order numerical schemes for the numerical

discretization of System (3.18). High order methods (Spectral type methods) have been

extensively used in computational fluid dynamics due to their accuracy and efficiency

in the simulation of fluid flows. In particular, these methods are suitable for problems

where high accuracy is required and, hence, are well suited to track the evolution of small

perturbations as in stability analysis.

In this context, a Spectral Discontinuous Galerkin method is used in this work to solve

equation (3.18). To do that, the original domain is divided into non-overlapping hexahedral

sub-domains, Ek, such that Ω =∑

k Ek. Inside each sub-domain, a polynomial of degree

N is used to approximate the unknowns and the fluxes, U,F, thus:

30 3.4 Three-dimensional Direct Numerical Simulation

UN =

N∑i,j,k=0

Ui,j,kΦi,j,k, FN =

N∑i,j,k=0

F(Ui,j,k)Φi,j,k, (3.19)

where

Φi,j,k = Li(x)Lj(y)Lk(z),

is the tensor product of the Lagrange interpolant in the nodes i, j, k and Ui,j,k is the value

of the unknown in each computational node. In this work, the nodes in each direction

follow a Gauss-Legendre distribution, and the basis functions L(i,j,k) are taken as the

Lagrange interpolant at these nodes.

Reconsidering equation (3.18), we obtain, at an element level, the following discretized

equation:

∂UN

∂t+∇ · FN = 0. (3.20)

The Discontinuous Galerkin-Spectral Element method (DG-SEM, Kopriva [59]) makes

use of the Galerkin weak form of the equations and a discontinuous treatment of the

interfaces and boundaries. Thus, equation (3.20) is multiplied by a test function (the

same function as the basis for the Galerkin method) and integrated in the computational

space, then the error is forced to be orthogonal at each test function Φi,j,k in a mesh

element Ek, yielding:

(∂UN

∂t,Φi,j,k

)Ek

+ (∇ · FN ,Φi,j,k)Ek= 0, (i, j, k) = 0...N,

with (a, b)Ek=

∫Ek

ab defining an inner product (typically the L2 inner product). After

integrating by parts we obtain:

(∂UN

∂t,Φi,j,k

)Ek

− (∇Φi,j,k, ·FN )Ek+

∫∂Ek

Φi,j,kFN · n �dS = 0, (3.21)

where the third term (the surface integral) extends over the boundary ∂Ek of the com-

putational element Ek, with external pointing normal n. This boundary may lie at the

interface between two elements or at a physical boundary conditions, and, in both cases,

the treatment is similar. Note that all integrals in equation (3.21) can be numerically

evaluated using Gauss quadrature.

To obtain a solution over the complete discretized computational domain (Ω =∑

k Ek)

it is necessary to sum all the element contributions:


(∂UN

∂t,Φi,j,k

)Ω

− (∇Φi,j,k, ·FN )Ω +∑γ∈Γ

∫γΦi,j,kF

∗(n,UL,UR) �dS = 0, (3.22)

where Γ denotes the set of internal edges in the mesh Ω. In addition, note that we

have replaced FN by F∗(n,UL,UR) in the surface integral. F∗(n,UL,UR) represents

the numerical flux between two consecutive elements in the mesh (Left and Right). This

numerical flux arises from the discontinuous Galerkin setting where we consider that each

element is disconnected from the next and hence contains a complete set of degrees of

freedom to represent a polynomial of order N.

Taking into account the decomposition of the unknown (equation 3.19) and the or-

thogonality of the Lagrange basis in the Gauss nodes, the following expression is finally

obtained for the integrals of equation (3.22).

∂Ui,j,k

∂t+DxF

1i,j,k +DyF

2i,j,k +DzF

3i,j,k = 0, (i, j, k) = 0...N. (3.23)

The discrete divergence (second term of the previous equation) is obtained after the

numerical integration of the second and third terms of equation (3.22). Gauss quadrature

is used to evaluate these integrals, giving:

DxF1i,j,k = F1∗(x, yj , zk)

Li(x)wi

∣∣∣x=1

x=0−

N∑m=0

F1m,j,kdi,m,

DyF2i,j,k = F2∗(xi, y, zk)

Lj(y)wj

∣∣∣y=1

y=0−

N∑m=0

F2i,m,kdj,m,

DzF3i,j,k = F3∗(xi, yj , z)

Lk(z)wk

∣∣∣z=1

z=0−

N∑m=0

F3i,j,mdk,m,

(3.24)

with:

dm,n = L′m(sn)wn

wm. (3.25)

In the previous expression, wn are the Gauss integration weights in dimension x, y, or z,

L′m(sn) is the derivative of the Lagrange interpolant evaluate in the node sn, and F∗ are

the interface fluxes. These fluxes can be differentiated into viscous or inviscid.

Computation of inviscid fluxes requires taking into account the left and right values

of the unknowns at each interface. Let us note that by taking the average value of the

unknowns U at the interface (equivalent to a central scheme) provides a numerically

unstable scheme when the convective terms dominate and is only recommended at very

low Reynolds numbers. For larger Reynolds numbers, an upwinding scheme should be used

32 3.5 Dynamic Mode Decomposition

instead. The most common way to introduce upwinding in the scheme is by solving the

equivalent Riemann problem at the interface. For the particular case of Euler equations

(or inviscid NS) different Riemann solver has already been developed. In this work, a

standard Roe Riemann solver has been used in the computations Toro [107].

The viscous fluxes require discretization for elliptic type equations. A simple approach

consists of averaging the right and left viscous fluxes at the interface, but this solution

has been proof numerically unstable for implicit schemes. A more general framework

for derivation and analysis of discontinuous Galerkin methods for elliptic equations (e.g.

Interior Penalty, Local Discontinuous Galerkin, Bassi-Rebay) was derived in Arnold et al.

[7]. Additional details on implementation, methodology and numerical validation of the

employed tool can be found in Jacobs et al. [53], Kopriva [59, 60].

3.5 Dynamic Mode Decomposition

The Dynamic Mode Decomposition (DMD) is a recent data-based technique, introduced

by Schmid Schmid [92], that follows the Koopman analysis of a dynamical system to

find the relevant spatial modes that evolve in a flow field, as previously discussed on the

introductory section (for additional details the reader is oriented to Rowley et al. [88] and

Mezic [67]). It is classified as a data-based technique because the only input required by

this post-processing method is a set of data snapshots, coming from numerical simulations

or experimental measurements. These flow field snapshots have to be collected with a

constant sampling frequency, dictated by the Nyquist criterion. Therefore, in order to

avoid aliasing and a diverged decomposition, the data must be sampled at least at twice

the highest frequency of the dynamic modes to be captured from the analyzed flow field.

In the present work, the sampling frequency is determined by the time each snapshot

has been saved, i.e. 10 non-dimensional time units. Therefore, the decomposition is only

able to capture features oscillating with a maximum StD of 0.05. A snapshot matrix can

then be constructed containing the selected N snapshots, temporally ordered and equally

spaced by the aforementioned constant sampling time:

V N1 = (v1, v2, v3, ..., vN ) (3.26)

The matrix V N1 may be composed of one or all variables of the flow field. If one con-

siders only one variable, the computational burden of the DMD is substantially reduced.

Nonetheless, to describe more accurately the whole system dynamics as much flow field


variables as possible are needed, given that the basis of the dynamic modes are common

to all of them Richecoeur et al. [80]. In the present chapter, it was verified that perform-

ing the DMD with the three velocity components of the flow or only with the spanwise

velocity component did not produce relevant differences on the obtained dynamic modes.

The results presented and discussed in the following section are the ones obtained from

the DMD using only a single variable (z-velocity component).

The fundamental idea of the DMD is to extract the dynamic characteristics of the linear

operator A that approximates the dynamical process between two consecutive snapshots,

as follows:

A(v1, v2, ..., vN−1) = (v2, v3, ..., vN )

AV N−11 = V N

2

(3.27)

For a sufficiently long sequence of snapshots, after performing a Singular Value Decom-

position (SVD) on the snapshot matrix V N−11 = UΣWH and projecting matrix A onto a

basis spanned by the POD modes U , we find a matrix S that can describe the dynamics

of the unknown matrix A, being computed as follows:

S = UHAU = UHV N2 WΣ−1 (3.28)

The following step consists on solving the eigenvalue problem Syi = μiyi. The dynamic

modes can finally be recovered using the following expression:

φi = Uyi (3.29)

From the phase and magnitude of the eigenvalues of S we retrieve the frequencies and

growth rates of the dynamic modes, respectively. For a complete description of the original

DMD algorithm, see Schmid [92].

4The three-dimensional dynamics

Contents

4.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 On complex conjugate eigenvalues . . . . . . . . . . . . . . . . . 39

4.2 Global instability validation . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Comparison with Bres & Colonius . . . . . . . . . . . . . . . . . 44

4.2.2 Comparison with 3D DNS . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Parametric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 Preliminary considerations on the envelope of stability . . . . . . 50

4.3.2 Spanwise wavenumber (β) . . . . . . . . . . . . . . . . . . . . . . 53

4.3.3 Reynolds number (ReD) . . . . . . . . . . . . . . . . . . . . . . 55

4.3.4 Incoming boundary-layer thickness (θ0/D) . . . . . . . . . . . . . 57

4.3.5 Aspect ratio (L/D) . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.6 Eigenmode morphology . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.7 Synthesis of the results . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.7.1 Incoming boundary-layer thickness and Reynolds number 71

4.3.7.2 Parametric dependence on aspect ratio . . . . . . . . . 72

4.4 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . 76

4.1 Problem description

A modal point of view is followed and BiGlobal instability analysis has been used to

analyse the flow over an open cavity. Lengths have been scaled with the depth of the

cavity, D, and two independent Reynolds numbers are used to characterise the flow, the

one based on the cavity depth, ReD = u∞D/ν, and the other based on the boundary layer

momentum thickness at the upstream cavity lip (see figure 4.1), Reθ0 = u∞θ0/ν. In the

vast majority of the cases, and unless otherwise stated the Reynolds is based on cavity

depth, and the value of incoming boundary-layer thickness (δ/D) have been chosen to

characterise the flow. The subscript on δ denotes the position in the streamwise direction

where the boundary-layer thickness is measured. So, δ0/D corresponds to the leading

35

36 4.1 Problem description

edge of the cavity while δ−1/D is the beginning of the computational domain (x−1),. The

distance between the inflow boundary and the upstream cavity lip, x0 − x−1 = D, was

always kept constant. As it was explained in the previous chapter, changing the inflow

condition allowed to control both the Reynolds number and the incoming boundary layer

thickness, without needing to change the domain.

L

INFL

OW

OU

TFLO

W

FARFIELD

WALL

-1

D

0x-1x

Figure 4.1: Schematic description of the two-dimensional open cavity and problem parameters.

Some of the linear analysis results presented in this work have been obtained fixing

the incoming displacement thickness of the boundary layer at the inflow boundary to

δ∗−1 = 0.25 (corresponding to momentum thickness θ−1 = 0.0337). The original reason for

this procedure was to reduce the degrees of freedom in the comparison with experimental

work and other authors’ numerical results. It would have been a better idea from a results

point of view to have a constant θ0, given that this parameter is the characteristic length

of the incoming boundary layer thickness that dominates the behavior, instead of using

a derivated length. On the other hand, the use of the boundary layer thickness at the

entry of the domain resulted in an ease of computation of the base flow results for the

cases of constant δ. Table 4.1 summarises the steady flow results obtained at different

Reynolds numbers, with constant incoming boundary layer thickness at the entry of the


domain; those at ReD = 1500 correspond to run ”2M01” in the work of reference by Bres

and Colonius [20] and will be discussed further in section 4.2. The change of θ0 with

the Reynolds numbers, when the value of θ−1 = 0.0337 is kept constant, can be seen in

figure 4.2. As it can be appreciated, the variation of boundary layer thickness is small,

especially for high Reynolds numbers.

Table 4.1: Parameters characterising the inflow Blasius boundary layer and associated location

of the primary, (xC , yC), and secondary, (xc, yc), vortices.

ReD x0 δ∗0 θ0 Reθ0 (xC , yC) (xc, yc)

1100 3.845 0.1017 0.0392 43.19 ( 1.408 , -0.412 ) ( 1.381 , -0.594 )

1300 4.363 0.0996 0.0384 50.01 ( 1.409 , -0.417 ) ( 1.388 , -0.566 )

1500 4.880 0.0981 0.0378 56.81 ( 1.410 , -0.420 ) ( 1.393 , -0.536 )

1700 5.397 0.0969 0.0374 63.61 ( 1.412 , -0.422 ) ( 1.397 , -0.515 )

1900 5.915 0.0960 0.0370 70.39 ( 1.414 , -0.425 ) ( 1.398 , -0.498 )

2400 7.207 0.0943 0.0363 87.33 ( 1.418 , -0.429 ) ( 1.400 , -0.473 )

0.02

0.025

0.03

0.035

0.04

0.045

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000ReD

0 /D

Figure 4.2: Variation with the Reynolds number of the incoming boundary layer thickness at

the leading edge of the cavity, θ0/D, for constant boundary layer thickness at the start of the

computational domain θ−1/D = 0.0337.


Figure 4.3: Schematic description of the 3-D open cavity and problem parameters.

In the complete parametric study presented in this chapter the effect of the variation

of the incoming boundary layer thickness is also studied.

Regarding the geometrical parameters, the two considered are the length-to-depth as-

pect ratio (L/D) and the wavelength in the spanwise direction (Lz) with the corresponding

wavenumber (β = 2π/Lz). A schematic description of the three dimensional configura-

tion is depicted in figure 4.3. The use of β as a parameter on the BiGlobal analysis was

explained in section 3.3. From a geometrical point of view β is more a characteristic of

the eigenmodes, like the frequency, in the sense that the eigenmodes have characteristics

lengths of maximum amplification, corresponding with the appropriate β, the same way

that they have a characteristic frequency. The nature of the analysis used here is such

as that β is a parameter of the problem, the value of β analyzed can be changed, and

performing a sweep on it the one of maximum amplification, and the limits of instability

of each eigenmode, can be determined. There are two main three-dimensional eigenmodes

for the open cavity problem, referred in what follows as Mode I (depicted in red in the

figures) and Mode II (in blue), or bifurcated mode. Mode I corresponds to a pair of com-

plex conjugate eigenvalues, fact that is usually referred in the literature as a travelling

mode. Mode II is a bifurcated mode, as it suffers a bifurcation along the β parameter. For

higher β the mode is stationary, while for lower β it also corresponds to a pair of complex

conjugate eigenvalues. The nature and behaviour of those two modes will be thoroughly

documented in this chapter.


4.1.1 On complex conjugate eigenvalues

The basis for the BiGlobal instability analysis was explained in section 3.3. If one would

decompose the flow into a sum of the base flow, plus a series of eigenmodes, the resulting

expression would be:

q(x, y, z, t) = q(x, y) +

n∑i=1

αiqi(x, y)ei βiz−i 2π

StDiU∞D

t+σit, (4.1)

where αi is the initial amplitude of each eigenmode, which is supposed to be very small

in the linear regime. In a real flow, each of those amplitudes would be a function of the

initial conditions, random noise in the flow, etc., but due to the exponential nature of the

grow of the eigenmodes, after a short period of time all the eigenmodes except the one of

greater amplification σ would be negligible. That mode would be, in this case, either part

of Mode I or Mode II, with the appropriate β of maximum amplification. In the case of a

real three-dimensional flow with non-periodic spanwise boundary conditions that β could

be selected somehow.

If the leading mode for a random flow condition is the stationary branch of Mode II,

then a good approximation of the flow in the growth regime would be:

q(x, y, z, t) = q(x, y) + αq(x, y)ei βz+σt, (4.2)

as in that case the Strouhal of the eigenmode is zero. In that case, there is only one

parameter for the composition, α, and given that there is no normalization conditions for

the eigenvector q(x, y) that constant can be set to one, and just use the time to describe

the growth of the mode. These stationary mode have a morphology that is characteristic,

since the function in the spanwise direction is simply periodic. When representing the

eigenmodes the isosurfaces of the w velocity component are represented. The spanwise

velocity component, w, is zero in the base flow (w = 0), so what is shown is only the

perturbation (w = w), whether it is a single eigenmode, or a pair of complex conjugates.

As an example, in figure 4.4 the stationary branch of Mode II is shown.

The other possible case is that the leading eigenmode is Mode I, that is, a mode with

a complex conjugate eigenvalue. The flow approximation would be, then:

q(x, y, z, t) = q(x, y) + α1q1(x, y)ei βz−i 2π

StDiU∞D

t+σt + α2q2(x, y)ei βz+i 2π

StDiU∞D

t+σt.

(4.3)


Figure 4.4: Spanwise velocity component, w, isosurfaces for a stationary eigenmode.

In this case the growthrate σ of the two eigenvalues are exactly the same, and there are

two relevant amplifications α1 and α2. There is no condition on the linear analysis that

establishes a value or a relationship between both amplifications. The problem, as stated,

has an additional degree of freedom, the way the two complex conjugate eigenvalues are

combined. In the case of a real flow, that degree of freedom would be determined by the

boundary or initial conditions. This kind of perturbations are usually called travelling

waves in the literature, and here it can be seen why, as the problem results in two waves,

one travelling left, and the other travelling right. Thus, different values of the constants

lead to different looking modes. In some cases, the combination of both eigenmodes forms

a standing pulsating perturbation, as the interference pattern of both waves cancels the

movement. In other cases, the combination of modes leads to structures that entwine,

and travel left or right, and the values of said combination affect the morphology of the

structure. Thus, when representing an eigenmode with complex conjugate eigenvalues

a value for the α-constants has to be chosen, to show a characteristic depiction of the

eigenmode. In figure 4.5 examples of different combinations of α are shown.


Figure 4.5: Spanwise velocity component isosurfaces for a travelling eigenmode for different

combinations of α1 and α2. On top, pulsating perturbation. The others are structures that are

right-travelling (middle row) or left-travelling (bottom).

42 4.2 Global instability validation

4.2 Global instability validation

Before addressing the stability analysis of the open cavity, the methodology employed has

been extensively validated in several test cases. Among these tests, the analyses performed

for confined cavity flow with cross-sectional geometries of increasing complexity de Vicente

et al. [29] have been a decisive tool for evaluating the domain decomposition methodology

and the accuracy of the resulting solution depending on different conditions, such as the

state of convergence of the basic flow, number of points in each computational domain,

or the number of vectors in the Krylov subspace base used in the iterative procedure for

recovering the leading eigenvalues of (2.36). Extensive validation of the instability analysis

algorithm has been performed. Previous works (de Vicente [27], de Vicente et al. [29])

have validated the domain decomposition methodology and the accuracy of the provided

solution.

The aim of this section is, then, to compare, as well as confront, the present results

with previous studies of rectangular cavity flows and with additional DNS computations

in order to check the validity of the BiGlobal linear analysis. Some of the results of the

BiGlobal analysis are briefly discussed here, but they will be studied in depth in section 4.3.

Two main cases are considered here, and also in Chapters 6 and 7.

The first case under consideration is Case A: ReD = 1500, whose details are provided

in table 4.2. The nominal conditions are those of Bres and Colonius [20]. The experi-

mental data (see Chapter 6) correspond to slightly different values for this case, with the

associated uncertainties shown in the table. The values of the linear analysis correspond

to a parametric sweep around the nominal conditions of this case (of both the experiments

and Bres and Colonius [20]), to take into account the uncertainties on the experimental

campaign. Unless otherwise stated the following comparisons have been made at nominal

conditions for each case.

Uncertainties on the flow conditions on experiments are critical when comparing results

with the linear analysis, as small variations of Reynolds number, or incoming boundary

layer thickness can have a huge effect on the stability of the flow. To illustrate that

effect, figure 4.6 shows the amplification rates of the leading modes in the critical range

of β for different ReD and θ0 in order to quantify the effect of varying each parameter

independently. Full symbols have been obtained keeping ReD constant (ReD=1500) while

θ0 varies from θ0 = 0.0379 to θ0 = 0.0475. Empty symbols, on the other hand, describe

the change in the amplification when ReD changes and θ0 is fixed. Being able to measure


Table 4.2: Parameters for Case A (ReD = 1500). B&C stands for the nominal conditions of

Bres & Colonius 08. ED for the Experimental Data of Chapter 6, with associated uncertainties.

LSA for the values of Linear Stability Analysis. The values of δ∗0 and θ0 are calculated with an

equivalent rig to the one employed in Chapter 6 for B&C and LSA.

δ∗0 (mm) θ0 (mm) θ0 =θ0D Reθ0 ReD

B & C 4.90 1.89 0.0378 56.8 1500

ED 5.59 ± 0.28 2.16 ± 0.11 0.0432 ± 0.0022 65 ± 4.9 1500 ± 43

LSA 4.42 - 8.87 1.71 - 3.43 0.0341 - 0.0685 51.2 - 102.9 1450 - 1550

4 6 8 10 12-0.01

-0.005

0

0.005

0.01

Re = 1500 = 0.0379Re = 1550 = 0.0432Re = 1500 = 0.0432Re = 1450 = 0.0432Re = 1500 = 0.0475

D

D

D

D

D

Figure 4.6: Effect of small variation of the flow conditions on the stability. Amplification vs

β, with full symbols corresponding to variations in incoming boundary layer thickness and empty

symbols to variations in ReD.

the sensitivity in the response to these two parameters is useful when comparing numerical

solutions with experiments. As it can be seen, variations of the parameter of the same

order of magnitude as the experimental uncertainties can have a big impact on the stability

of the flow. The reference value: ReD = 1500 and θ0 = 0.0432 (triangles) corresponds


to the nominal parameters of the first experimental case. Full circles, on the other hand,

corresponds to the numerical computations at ReD = 1500 and θ0 = 0.0379. An important

remark must be made at this point, the second mode to become unstable shows a big

dependence on the aforementioned flow parameters, this effect, in addition to the rather

wide margin of uncertainty in the experimental flow conditions, could affect the comparison

between both approaches, numerical and experimental.

The parametric space in the BiGlobal analysis of this case has been defined in a wide

range (see table 4.2), to both comprise the experimental conditions and also to determine

the critical instability parameters. As it has been already stated, a small variation on ReD

was considered to deal with the uncertainty in the experimental flow conditions (figure 4.6).

The nominal values of Case B are presented in table 4.3. In this case, there is no

additional information in the literature to compare with, so all information related to this

case and the experimental campaign is presented in Chapter 6. Also, in this case the values

presented for the Linear Stability Analysis are just the nominal ones, as no study of small

variation of parameters was performed, equivalent the one in figure 4.6, and the value of

the incoming boundary layer thickness used for the nominal case is slightly different, as

can be seen in the table. That effect can be nevertheless considered in light of the results

presented in the rest of this chapter, and the difference with the experimental results due

to incoming boundary layer thickness should be negligible, given that the flow regime is

unstable, and very far away from the neutral curve.

Table 4.3: Parameters of the nominal conditions for Case B (ReD = 2400).

θ0 =θ0D Reθ0 ReD

Experimental Data 0.0340 ± 0.0015 81 ± 5.0 2400 ± 51

Linear Stability Analysis 0.036 84.6 2400

4.2.1 Comparison with Bres & Colonius

The constant value of θ−1 chosen allows a straightforward connection with the results

provided by Bres and Colonius [20]. This work is to-date the most complete account

of compressible flows instability over open cavities. In run 2M01 the authors detail the

stability characteristics of compressible flow for an open cavity with aspect ratio L/D = 2

with Mach number M= 0.1, ReD = 1500 and Reθ0 = 56.81. The low Mach number permits


the comparison with the incompressible limit performed here. The (single) most unstable

mode obtained by Bres and Colonius [20] lies inside the unstable region, almost along the

line of maximal amplification predicted by the present analysis.

As it can be seen in figure 4.7 the four points provided in that work, corresponding

to the three leading modes, match quite well both in frequency and amplification with

the results here presented. In figure 4.7 the results are presented in the same way as it

Figure 4.7: Comparison between Bres and Colonius [20] results (black diamonds) and present

BiGlobal analysis solution (Mode I in red circles, Mode II (bifurcated) in blue circles) for the

nominal conditions of case A. Amplification rate as a function of the spanwise length on the left

and of the dimensionless frequency on the right.

was in Bres and Colonius [20], using the spanwise length of the perturbation (as always

dimensionalized with the depth of the cavity), Lz to describe it. The authors identified

three different modes as the most unstable or least stable for different ranges of β = 2π/Lz,

one of them being indeed unstable. The results disclosed in the present work, using a finer

discretization on the β parameter, show that, for the chosen flow parameters, there is

another unstable mode, this one stationary, for β 10. The mode can be identified in

figure 4.7 (left) where a peak in the unstable region appears at Lz 0.62. Its stationary

nature is certified in figure 4.7 (right) where the peak arises when ωi = 0. It is also

important to address that two of the modes identified by Bres and Colonius [20] correspond

with the stationary and travelling parts of the bifurcated mode, as it will be explained

with greater detail in the next sections, as the rest of this chapter delves more deeply in

the nature of those eigenvalues.


In figure 4.8 the same results are plotted, this time using the parameter β, which is

the one employed in the rest of this thesis.

Figure 4.8: Amplification versus β for Case A. Mode I in red circles, Mode II (bifurcated) in

blue circles and Bres and Colonius [20] results in black rhombi.

The Strouhal number corresponding to the dominant oscillation frequency, in run

2M01, is St= 0.025. The spanwise wavelength of leading eigenmode is also reported and

its value is β = 1. In the present analysis, the amplification rate of the leading eigenmode

predicted is σ = 0.00981 while the frequency is ω = 0.1623, leading to a Strouhal number of

St= 0.0259. For visualisation purposes the isosurfaces of the spanwise velocity component

(w) are shown in figure 4.10 (left), in which the three dimensional spanwise disturbance

is reconstructed periodically attending to the characteristic wavenumber β = 6, showing

two characteristic lengths in the domain Lz = 2.1.


4.2.2 Comparison with 3D DNS

Another numerical comparison in the cases A and B has been done using the three-

dimensional non-steady DNS explained in section 3.4. The comparison is expected to be

exact in the linear regime of the DNS.

The DNS computation has been designed acording to the information provided in

table 4.2 and table 4.3. A Mach number equal to 0.1 has been selected, minimizing the

compressibility effects. Periodic boundary conditions in the spanwise direction have been

used, fixing a periodicity length of Lz = 1.05 corresponding to a real spanwise wavenumber

β = 6.

After a long transient, due to the small amplification rate of the leading disturbance,

a periodic behavior with exponential growth is reached. In figure 4.9 the L∞-norm of

the spanwise velocity component of the perturbed flow is monitored as a function of non-

dimensional time. The exponential growth is obtained by a linear least squares technique

fitting the maximum of each period, while a Fast Fourier Transform (FFT) is accomplished

to extract the frequency. As happened in the comparison with the Bres & Colonius

work there is an excellent agreement between DNS and linear analysis results using the

BiGloblal methodology when the wavenumber in the spanwise direction is fixed β = 6. The

amplification rate of the leading eigenmode predicted σ = 0.00981 matches with the one

obtained by DNS σ = 0.00978 while frequencies show an analogous conduct: ωi = 0.1623

vs ωi = 0.1634 computed from DNS results. This results, as well as the ones corresponding

with the next case can be seen in table 4.4.

Finally in figure 4.10 the isosurfaces of the spanwise velocity component (w) of the lead-

ing travelling disturbance are shown for both the DNS computation and the BiGlobal anal-

ysis. The three dimensional spanwise velocity disturbance, left picture, is reconstructed

periodically attending to the characteristic wavenumber β = 6, showing two characteristic

lengths in the domain Lz = 2.1. DNS solution, right figure, reveals a qualitative similar

appearance.

These structures are in good agreement with those obtained in previous analyses, not

only that of Bres and Colonius [20] but also the earlier analysis of Theofilis and Colonius

[104] (in which only the domain inside the cavity was analysed) and that of Theofilis and

Colonius [103], who employed the residuals algorithm from Theofilis [100], all such results

being presumably related with the wake mode instability.


non-dimensional time

log

(m

ax|

|)

400 600 800 1000

10-11

10-10

10-9

w

Figure 4.9: Temporal evolution of the maximum value of the spanwise velocity perturbation

obtained by DNS.

Figure 4.10: 3D Visualization of spanwise velocity at ReD = 1500 and β = 6: Leading disturbance

obtained using BiGlobal analysis on the left; DNS solution on the right.

Second case, Case B: ReD = 2400, exhibits a more challenging behaviour. The first

drawback to consider comes from the fact that the base flow employed in the computations

is artificially stabilised due to the nature of the numerical method used. The effects of


using this kind of mean flow in the stability analysis have been previously studied in

Barkley [11] for wake flows and in Piot et al. [78], where a good agreement between the

simulations and linear stability analysis results was observed.

BiGlobal results confirm those findings, two unstable two-dimensional perturbations

are recovered: a mode with frequency StI 0.257 and other with frequency StI 0.42.

However, in the parametric space studied, the flow exhibits three-dimensional disturbances

with higher amplification rates, more relevant in the framework of this work.

According to linear analysis results, as it will be explained in the next section, in the

range of β between β 2 and β 18, the predicted numerical solution is a combination

of three unstable modes. The most unstable one, see figure 4.15, is one of the stationary

(ωi = 0) branches in which Mode II splits. The amplification rate of this disturbance

reaches its maximum σ = 0.044 for β 11.7.

Figure 4.11 depicts the isosurfaces of the three dimensional reconstruction of the span-

wise velocity component (w) of the leading growing disturbance obtained by linear analysis.

Two characteristic length are shown in the domain Lz = 1.05. The structures presented

here are morphologically similar to the ones seen in Faure et al. [36]. In that work, the

aspect ratio was fixed to L/D = 1, but the most unstable structure was the same bifur-

cated mode of β = 12 (see figure 4.23). Once more, there is great agreement between the

BiGlobal results and the linear part of the DNS (see table 4.4). Additional results on the

non-linear part of the DNS computations can be seen in Chapter 7.

Figure 4.11: 3D Visualization of spanwise velocity disturbance obtained at ReD = 2400 using

BiGlobal analysis (left) and DNS at t = 400 (right).

50 4.3 Parametric analysis

Table 4.4: Amplification rate (σ) and frequency (StD) of the leading BiGlobal mode and the

linear part of the DNS, for both cases.

Case Method StD σ

Case A BiGlobal 0.0258 0.00981

Case A DNS 0.0260 0.00978

Case B BiGlobal 0 0.0439

Case B DNS 0 0.0440

4.3 Parametric analysis

The parametric dependence of the stability features of the flow is now discussed in a three-

dimensional framework (β �= 0). Besides the geometric parameter aspect ratio L/D, the

wavelength in the spanwise direction, β, is also examined. In addition, the effect of the

incoming boundary-layer thickness and the Reynolds number has been investigated in this

section. Table 4.5 contains the ranges of the different parameters examined.

Table 4.5: Range of the parameters studied

L/D 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 2 and 3

ReD ∼ 800 to 4600

β ∼ 0 to 22

δ0/D ∼ 0.05 to 0.3

4.3.1 Preliminary considerations on the envelope of stability

The separation between the stable and unstable regimes in the parametric space is a

hypersurface. This manifold marks the change of stability of the critical eigenvalue at each

point of the parameter space. Owing to its multi-dimensional nature, the representation

of that hypersurface is rather complex, so in order to characterise the stability boundaries

neutral curves are presented as cuts through the hypersurface with parameter planes,

namely ReD vs β, or ReD vs θ0/D. In the next sections, the effect of Reynolds number

and spanwise length on the stability features of the eigenmodes is presented. Subsequently

the consequence of varying the incoming boundary-layer thickness θ0/D and aspect ratio


L/D is analysed. To conclude this chapter a synthesis of the results, including the relevant

parametric dependencies, is presented.

The most significant result obtained using the BiGlobal analysis is the identification

of the critical parameters related to the destabilisation of the flow over the open cavity.

To illustrate this behaviour is useful to map the different regions in the parametric space

depending on the nature of the flow. Not only that, within the unstable region of the pa-

rameter space, linear analysis can give information about which modes are more unstable,

and what are their characteristics.

The determination of these maps, also known as neutral curves, is done through solving

iteratively the eigenvalue problem, changing the relevant parameters, appropriately. The

effect of the different parameters will be explained in detail in what follows, but here

the analysis start with the variation of the spanwise wavenumber, β. In the left part of

figure 4.12 the relation between the frequency, StD, and the amplification rate, σ, of the

leading modes is shown for different β values, while all the other parameters (Reynolds

number, incoming boundary layer thickness and length do depth aspect ratio) are kept

constant. As can be seen the two critical eigenvalues for β = 4.4 (red) become more

stable as β increases, while the other two leading eigenvalues become unstable. In order

to determine the neutral curves, β has been changed in small steps, ranging from Δβ = 0.2

in zones where no interesting phenomena occurred to Δβ = 0.001 when trying to describe

the bifurcations in detail or to determine the values for which the change in stability

occurs.

The representation of the variation of amplification or frequency with β leads to graph-

ics like figure 4.7 and figure 4.8 shown in section 4.2.1. Changing other parameter, like,

for example, the Reynolds number, produces changes in those curves, making them more

unstable, or more stable, as can be seen in the right part of figure 4.12 (or previously

shown in figure 4.6). The black line in σ = 0 of figure 4.12 constitutes the limit of sta-

bility, where an eigenvalue changes its nature from stable to unstable. Or, looking at a

three-dimensional representation of the eigenvalues in amplification, σ, versus β and ReD

(as can be seen in figure 4.13) for a given L/D and θ0 the plane of σ = 0 marks that limit,

in this case the L/D = 2 neutral curves.

The neutral curves corresponding to the first three cavity eigenmodes are presented

in figure 4.14 for the L/D = 2 case. As it was previously mentioned, the value of the

momentum thickness of the incoming boundary layer at the inflow of the computational

domain is fixed θ−1 = 0.0337, while in the upstream lip of the cavity Reθ0 varies from


Figure 4.12: Left:variation of the eigenvalues amplification and frequency with β. L/D = 2,

Re= 2300, θ−1 = 0.0337, β = 4.4 in red, β = 4.8 in green, β = 5.2 in blue and β = 5.6 in black.

Right: Variation of the curves σ versus β with the Reynolds number. All cases are for L/D = 2

and θ−1 = 0.0337, Re= 2300 in red, Re= 2400 in green, Re= 2500 in blue and Re= 2600 in black.

Reθ0 = 43.19 to Reθ0 = 87.33 when ReD increases from the very stable ReD = 1100 to

the unstable ReD = 2900. The fine parametric scan employed (ΔReD 1, Δβ 0.01

in the nose of the modes) permits a precise identification of the three-dimensional critical

conditions for global stability analysis of this case, which are presented in table 4.6.

Table 4.6: Critical parameters of the first three modes for the open cavity flow with aspect ratio

L/D = 2 and θ−1 = 0.0337.

Mode Recrit βcrit

I 1149 5.62

II (a) 1471 9.86

II (b) 1522 4.45

Third 2207 10.34

Thus, the use of representations like figure 4.14 provides additional interesting infor-

mation related to the global stability of the cavity flow. It can be stated that below

ReD = 1149 the basic flow is three-dimensional globally stable for that L/D and θ−1. As

ReD increases, unstable perturbations, initially confined inside the cavity, begin to appear.

The characteristics of these leading eigenmodes: I, II and third in table 4.6, are explained

in detail in the next section. More information can be extracted from the data contained

in figure 4.13. Looking at the maximal amplification of the different eigenmodes, and how


Figure 4.13: Amplification, σ, versus β and ReD of the three leading modes with positive am-

plification in the open cavity flow of aspect ratio L/D = 2 with θ−1 = 0.0337. Mode I in red,

Mode II in blue and the third mode in white.

they change with the Reynolds number allows to identify which eigenmode is the lead-

ing disturbance for different flow conditions. This can be seen in figure 4.15 where those

maxima are represented for the first three modes.

4.3.2 Spanwise wavenumber (β)

In an idealised spanwise infinite cavity the first structures to amplify in a linear context

would be the ones that correspond to the most unstable for that given point in the param-

eter space. The spanwise wavenumber, β, would characterise the instability modes instead

of being a parameter of the problem. When dealing with three-dimensional centrifugal

instabilities two characteristic spanwise lengths for the periodic instabilities are found to

be inherently more unstable. Perturbations that are of the size of the cavity depth, β 6,


Figure 4.14: Neutral stability curves of the three leading modes in the open cavity flow of aspect

ratio L/D = 2 with θ−1 = 0.0337. Mode I in red, Mode II in blue and the third mode in white.

Critical values are cited in table 4.6.

and also perturbations with a spanwise length of about one half of the cavity depth, or

slightly bigger β 10 − 12. In addition, the two-dimensional instabilities, detailed in

Chapter 5, also tend to appear as a third main inherently unstable area in the parame-

ter space. In figure 4.16 the values of maximum amplification on the Reynolds versus β

appear superimposed to the neutral curves already shown on figure 4.14 (the precision of

the values on β is only of 0.2, as they were obtained with automatic post-processing of

the data, and correspond to zones in with the grid variation of β is coarse, without use of

interpolation).


Figure 4.15: Dependence on Reynolds number of the amplification rate, σ, of the leading eigen-

modes in the open cavity flow of aspect ratio L/D = 2 with θ−1 = 0.0337. Mode I in red, Mode II

in blue and the third mode in white.

4.3.3 Reynolds number (ReD)

The effect of Reynolds number on the flow stability has also been studied, and partially

explained in section 4.3.1. In general, as ReD increases so does the amplification/damping

rate (denominated σ) for all the eigenvalues, as increasing ReD means increasing the

velocity gradient at the shear layer, which is one of the main mechanisms for introducing

energy into the small perturbations.

The neutral curves for L/D = 2 were shown in figure 4.14. There are three main

unstable modes. The first one to become unstable (Mode I) for this configuration, in red

circles in the figures, is the mode associated with a pair of complex conjugated eigenvalues,

known in the literature as a travelling mode. The second mode, in blue circles, undergoes

a bifurcation with the variation of β. For low wavenumbers, there are two complex conju-

gated eigenvalues (so it is also a travelling mode) but when β grows those two eigenvalues


Figure 4.16: Neutral stability curves of the two leading modes in the open cavity flow of aspect

ratio L/D = 2 with θ−1 = 0.0337 in circles, approximate maxima in rhombus. Mode I in red,

Mode II in blue.

collapse and become two real eigenvalues, one of which gets stabilised, and the other one

gets destabilised. This disturbance will be named Mode II, or Bifurcated Mode in what

follows. The third unstable eigenmode shown, in empty circles, is also a travelling distur-

bance. That third mode is of little relevance and it is only present for relatively high ReD

configurations, in which the validity of the linear approximation can be questionable.

As it was shown in figure 4.15, the different modes change their amplification with

the Reynolds numbers at different rate, initially Mode I is dominant but as the Reynolds

number increases the unstable stationary branch of Mode II becomes dominant.


4.3.4 Incoming boundary-layer thickness (θ0/D)

To perform the study on the effect of the incoming boundary layer thickness the critical

parameter ReD and β of these modes have been tracked with the change of θ0. In figure 4.17

the behaviour of the critical ReD when θ0 varies from θ0 = 0.0317 to θ0 = 0.0665 is shown.

The critical wavenumber β remains almost constant (β = 5.62 for the leading and β = 9.86

for the stationary mode) independently of the boundary layer thickness, this effect is the

same when the flow parameter modified is ReD. To clarify the meaning of this figure, the

particular point (ReD, θ0) = (1149, 0.039) has been highlighted. This value corresponds to

the nose of the most unstable mode in figure 4.14. The parametric sweep in the ReD vs

θ0/D space is presented for Mode I and Mode II for L/D = 2, and Mode I for L/D = 3.

Both increasing ReD and decreasing boundary-layer thickness leads to larger gradients,

and eventually to destabilisation of the flow. The effect of decreasing the boundary-layer

thickness is qualitatively similar to increasing ReD for the three-dimensional instabilities,

while it has a much more pronounced effect in the two-dimensional shear layer modes.

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

700 1200 1700 2200 2700ReD

0 /D

Figure 4.17: Neutral curves in ReD vs θ0/D for the critical β. Mode I in red symbols, Mode II

(bifurcated) in blue symbols. L/D = 2 in circles and L/D = 3 in squares. Highlighted point

corresponds with the nose of the most unstable mode in figure 4.14.


4.3.5 Aspect ratio (L/D)

The effect of the aspect ratio on the stability is studied next and the dependence between

the size of the perturbations and the critical Reynolds number based on D is documented

(see Section 4.3.7). While L D the size of that perturbation is constrained by the

length of the cavity. Once L/D is big enough the vortex over which the perturbations

develop is not bounded by the cavity length. That means that the size of the perturbation,

denoted LP /D, is determined by the size of the recirculating vortex, not directly by L/D.

In addition, the different behaviour of the different modes with L/D leads to significant

differences in the neutral curves for each aspect ratio, as can be seen in figures 4.18, 4.19

and 4.20. Figure 4.18 shows just the evolution of Mode I with the L/D parameter, and

figure 4.19 does the same for Mode II, while figure 4.20 shows the neutral curves for each

of the L/D configurations studied.

Figure 4.18: Neutral curves for θ−1/D = 0.0337 of Mode I. Aspect ratio varying from L/D = 1

(higher ReD numbers) to L/D = 3 (lower ReD numbers) with the values detailed in Table 4.5.


Mode I presents a lobe with β ∼ 12 for low L/D (figure 4.18). In the case of L/D = 1

this is the second mode to become unstable (figure 4.20). When the aspect ratio increases

up to L/D = 1.1 a new lobe of that mode becomes unstable in the range of Reynolds

numbers studied, with a maximum amplification on the range of β ∼ 6. For the L/D = 1.2

cavity there is a change in the leading instability of this mode, as the low β lobe is the

first one to become unstable. Increasing the aspect ratio even more to L/D = 1.3 leads to

the main shift in stability, as Mode I becomes the first one to become unstable, while the

second lobe of that mode, now on β ∼ 11 starts to vanish (figure 4.20. By L/D = 1.4 the

mode comprises a single lobe with a maximum in the range of β ∼ 6, and a slight lump

of increased instability around β ∼ 11. As L/D continues to increase, to 1.5, 2 and 3 the

basic shape of the instability curves of the mode remains unmodified and only a change

in the rate of variation of the critical ReD can be appreciated, as it will be discussed in

the next section.

Figure 4.19: Neutral curves for θ−1/D = 0.0337 of Mode II. Aspect ratio varying from L/D = 1

(higher ReD numbers) to L/D = 3 (lower ReD numbers) with the values detailed in Table 4.5.


Mode II (figure 4.19) has an even more interesting behaviour, being a bifurcated mode.

At lower β a complex conjugate pair of eigenvalues is found in the analysis. Around β ∼ 9

a bifurcation occurs, and two real modes appear, one becoming more unstable (σ in-

creases), and the other one more stable, as σ decreases. So for L/D = 1 the most unstable

eigenmode corresponds to the real branch of the bifurcated mode, with a maximum am-

plification around β ∼ 12, i.e. a characteristic periodicity length around D/2. In the case

of this mode the most unstable part is always the real branch of high β. The difference

in critical Reynolds number between the two branches decreases as L/D increases. This

mode is the critical one for L/D = 1.2 and lower, as was discussed above when explaining

the features of Mode I. At L/D = 1.3 and L/D = 2 there is a small increase of instability

in the zone of β ∼ 3 and an small lobe is formed. In the case of L/D = 1.4 and L/D = 1.5

there is a turning point in that region, but a complete lobe is not formed. That change of

behaviour is interesting, because the trend is not continuous with the change of the L/D

parameter. Also, the only instance of critical ReD increasing as L/D increases can be seen

in the travelling branch of this mode, for L/D = 3.

Lastly, in figure 4.20 two other unstable modes can be seen in the range of parameters

studied. The third mode corresponds with a complex conjugate pair of eigenvalues, and it

can be seen between L/D = 1.2 and L/D = 2 with a maximum amplification at β ∼ 11.

This mode is more complex in structure, but it is relatively not very important for the

physical behavior of the flow. By the time this mode starts to become unstable the other

two modes have been developing out of the linear regime for any real flow, with saturation

or non-linear interaction of the modes, and even more, the relatively high ReD at which

it appears may imply that there are two dimensional instabilities present already, as it

will be explained in the two dimensional section of this paper. The last mode present is

a particularity of L/D = 3, being strikingly different, and appearing for β ∼ 3, where no

previous disturbances have been founded for lower cavity aspect ratios.

The variation of the critical Reynolds number of the two different lobes (when present)

of Modes I and II with L/D for the constant δ−1/D is shown in figure 4.21.

A summary of the main characteristics of the critical values when a change in stability

occurs for constant δ−1/D is shown in table 4.7. This table serves as a reference for the

rest of the figures, as it is impossible to provide numerical values for the whole problem

(i.e., when the variation of incoming boundary layer thickness is taken into account).


Figure 4.20: Neutral curves for θ−1 = 0.0337. From left to right, and from top to bottom,

L/D = 1.0, L/D = 1.1, L/D = 1.2, L/D = 1.3, L/D = 1.4, L/D = 1.5, L/D = 2.0 and

L/D = 3.0. Mode I in red, Mode II (bifurcated) in blue, third mode in white and fourth mode in

black.


0.81

1.21.41.61.82

2.22.42.62.83

800 1200 1600 2000 2400 2800 3200 3600 400ReD

L/D

Figure 4.21: Neutral curves in the ReD vs L/D for the critical β and θ−1 = 0.0337. First mode

in red, second mode in blue, with circles being low β and rhombi high β.

Table 4.7: Critical parameters of the first three modes for each of the aspect ratios and for

constant δ−1/D.

First Mode Second Mode Third Mode

L/D ReD β ReD β ReD β ReD β ReD β

1.0 - - 3937 13.47 - - 3421 13.07 - -

1.1 3630 6.56 3167 12.45 3884 6.63 2807 12.22 - -

1.2 2565 6.38 2659 11.60 3205 6.16 2383 11.54 3476 11.61

1.3 1985 6.23 2289 10.66 2608 5.63 2069 10.95 3071 11.07

1.4 1659 6.09 - - 2147 5.13 1849 10.45 2863 10.88

1.5 1472 5.96 - - 1853 4.84 1708 10.11 2724 10.83

2.0 1150 5.62 - - 1523 4.45 1471 9.86 2207 10.34

3.0 865 5.30 - - - - 1342 10.06 - -


4.3.6 Eigenmode morphology

For visualization purposes the structures corresponding with the w velocity component of

Mode I for different L/D can be seen in figure 4.22. It is interesting to notice the difference

in the characteristic lengths of the structures in the spanwise direction, corresponding

with the two different β lobes that appear for lower L/D configurations (L/D = 1.1 in the

figure). As it was previously said, the high beta lobe disappears for L/D > 1.3. Also, it is

remarkable how the structures are constrained by the left wall of the cavity. In the most

extreme case, for L/D = 3, the mode is located only in the downstream part of the cavity,

and does not occupy it in its entirety, which has an important effect in the stability, as it

will be explained in the next section.

In figure 4.23 the w velocity component for the stationary branch of Mode II with

different aspect ratios L/D are shown. As it happened with Mode I, for different L/D

the mode shows different morphology, as the upstream lobe position is affected by the

presence of the left wall of the cavity. In this case the lobe is straight for L/D > 1.5,

instead of being bent down, like it was in all the other cases. Several other interesting

morphological variations on the structure surrounding the main recirculating vortex can

also be observed, as the complexity increases with L/D.

In figure 4.24 the structures of the bifurcated mode at the other side of the bifurcation,

the low β branch, are shown. These structures correspond with the travelling part of the

mode. The aspect ratios are in the same position in figure 4.24 as they were in figure 4.23,

to make easier the comparison between both (except of the top left one, corresponding to

L/D = 1.1 instead of L/D = 1, given that in the range of Reynolds number studied there

is no low β lobe for L/D = 1). A very remarkable change in behavior occurs in this case,

as the mode for larger L/D extends to the secondary recirculating vortex, occupying the

whole cavity, even for L/D = 3. That anomaly might explain why in that case the critical

Reynolds number of that branch is higher for L/D = 3 than for L/D = 2. In the next

section the relationship between the length of the perturbation and the critical Reynolds

number is explained, and in this case the radical change of shape of the eigenmode affects

such relation.

Finally, the last, and less relevant, eigenmodes are shown in figure 4.25. The fourth

mode, on top, has a high β, and shows a more complex coiling than Mode I. The fourth

mode, only present in L/D = 3 shows a radically different structure, unlike any of the

other modes, not only occupying the whole cavity, but having a maximum amplification

in the range of β 3, lower than any other mode.


Figure 4.22: From left to right and top to bottom, w velocity component isosurfaces of Mode I

for, L/D = 1.1 (ReD = 3167, β = 12.45), L/D = 1.1 (ReD = 3630, β = 6.56), L/D = 1.5

(ReD = 1472, β = 5.96), L/D = 2 (ReD = 1150, β = 5.62) and L/D = 3 (ReD = 865, β = 5.30)

respectively. In all cases the spanwise length shown is Lz = 2π/β = 2D.


Figure 4.23: From left to right and top to bottom, w velocity component isosurfaces for the

stationary branch of Mode II for, L/D = 1 (ReD = 3421, β = 13.07), L/D = 1.3 (ReD = 2069,

β = 10.95), L/D = 1.5 (ReD = 1708, β = 10.11), L/D = 2 (ReD = 1471, β = 9.86) and L/D = 3

(ReD = 1342, β = 10.06) respectively. In all cases the spanwise length shown is Lz = 2π/β = 2D.


Figure 4.24: From left to right and top to bottom, w velocity component isosurfaces for the

travelling branch of Mode II for, L/D = 1.1 (ReD = 3884, β = 6.63), L/D = 1.3 (ReD = 2608,

β = 5.63), L/D = 1.5 (ReD = 1853, β = 4.84), L/D = 2 (ReD = 1523, β = 4.45) and L/D = 3

(ReD = 1638, β = 4.61) respectively. In all cases the spanwise length shown is Lz = 2π/β = 2D.


Figure 4.25: From left to right and top to bottom, w velocity component isosurfaces for the third

mode, L/D = 1.2 (ReD = 3476, β = 11.61), L/D = 2 (ReD = 2207, β = 10.34) and the fourth

mode of L/D = 3 (ReD = 1442, β = 2.92) respectively. In all cases the spanwise length shown is

Lz = 2π/β = 2D.

All the figures presented until now correspond to the w velocity component. In order

to visualize the process of mode growth figure 4.26, figure 4.27 and figure 4.28 show the

evolution of the three velocity components of the composition of the base flow plus the

travelling disturbance that is the peak of Mode I. Each figure represents one of the velocity

components at different times during the temporal evolution in the linear regime. In the

case of figure 4.26, showing the streamwise velocity component, the isosurfaces are shown

upside down to the usual, so the structure in the main recirculation vortex is not covered

by the ones of the developing boundary layer.


Figure 4.26: From left to right and top to bottom, u velocity component isosurfaces for Mode I.

Time with respect of the final time t/tf = [0; 0.25; 0.5; 0.625; 0.75; 0.875; 0.94; 1] In all cases the

spanwise length shown is Lz = 2π/β = 2D.

In the first two figures the recirculation vortex can be clearly seen, and also how the

perturbation starts to grow around it, varying its shape, and finally coiling around it. In

figure 4.28 the evolution of the spanwise velocity isosurfaces is shown. Since the times

shown are the same as in the last two figures, only the last four frames are shown, since

the value of w is too small to have any isosurface at t/tf < 0.6. These last three figures

allow the reader to have an idea of how does the eigenmode growth look like.


Figure 4.27: From left to right and top to bottom, v velocity component isosurfaces for Mode I.

Time with respect of the final time t/tf = [0; 0.25; 0.5; 0.625; 0.75; 0.875; 0.94; 1] In all cases the

spanwise length shown is Lz = 2π/β = 2D.


Figure 4.28: From left to right and top to bottom, w velocity component isosurfaces for Mode I.

Time with respect of the final time t/tf = [0.75; 0.875; 0.94; 1] In all cases the spanwise length

shown is Lz = 2π/β = 2D.


4.3.7 Synthesis of the results

In this section a description of relations between parameters of the flow and its global

instability characteristics is presented. An empirical law is provided when possible.

4.3.7.1 Incoming boundary-layer thickness and Reynolds number

In figure 4.17 it can be appreciated that for sufficiently thick boundary layers there seems to

exist a linear relation between the critical ReD and the incoming boundary-layer thickness

for the modes studied. Mode I of L/D = 3 presents a saturation regime for thin boundary

layers in which that linear behaviour does not occur. The slope of the linear part of the

curve depends on the length of which the Reynolds numbers is based. Basing Re on L

instead of D causes the neutral curves of Mode I for both L/D = 2 and L/D = 3 pictured

in figure 4.17 to have almost the same slope in the higher Reynolds range, as can be seen

in figure 4.29.

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1700 2200 2700 3200 3700 4200 4700 5200ReL

0 /D

Figure 4.29: Neutral curves in ReL vs θ0/D for the critical β. Mode I in red symbols, Mode II

(bifurcated) in blue symbols. L/D = 2 in circles and L/D = 3 in squares.


4.3.7.2 Parametric dependence on aspect ratio

The results of figure 4.21 indicate that an exponential fit can be constructed between the

critical ReD of the modes and the inverse of the aspect ratio, that is, D/L, for aspect

ratios lower than L/D = 1.5. That exponential law can be seen in figure 4.31 (top).

To explain that change in behaviour for higher aspect ratios, figure 4.30 contains a cut

of the spanwise velocity component of the low β lobe of Mode I for L/D = 1.2, L/D = 1.5,

L/D = 2 and L/D = 3, similar to what could be seen in figure 4.22. When L/D increases

sufficiently the size of the perturbation stops being constrained by the upstream wall

and it is more appropriate to use the characteristic length of the perturbation LP /D on

the flowstream direction in the aforementioned exponential law, as seen in figure 4.31

(top). That perturbation length can be defined, to use as reference, as the horizontal

distance between the 50% of the leftmost peak of the spanwise velocity component of the

perturbation and the downstream wall of the cavity.

ReDcrit = C1eC2D/LP . (4.4)

The parameters C1 and C2 for the two lobes of each of the first and second modes can

be found in table 4.8 (left).

Figure 4.30: Qualitative representation of the spanwise velocity component of the low β lobe of

Mode I for L/D = 1.2, 1.5, 2 and 3, from top to bottom and left to right.


800

1300

1800

2300

2800

3300

3800

1 1.5 2 2.5 3

ReD

L/D

4

5

6

7

8

9

10

11

12

13

14

1 1.5 2 2.5 3L/D; L P /D

Figure 4.31: Dependence with L/D and LP /D of the critical ReD for the critical β (top) and

said β of maximum amplification (bottom). All data with θ−1/D = 0.0337. Mode I in red symbols,

Mode II in blue symbols, with circles for low β and rhombi high β. In lines, the predicted curves

of equation 4.4, and the hyperbolic law described in § 4.3.7, and in small empty symbols, of same

shape and color, the equivalent LP /D of the same points.


Similar behaviour occurs in the case of the β of maximum amplification. While the

modes are in the range in which LP /D L/D there is a linear relation between L/D

and Lz (that is, an hyperbolic relation with β), as can be seen in figure 4.31 (bottom).

Out of that range, substituting the characteristic length for LP /D, yields that the curve

is being followed for Mode I. In the case of Mode II the behaviour is as expected for the

low β branch, except for the L/D = 3 point, that having a complete different morphology

does not follow it. The other branch has the same behaviour, while the morphology of

the modes is similar (see figure 4.23), the law is followed, but when the leftmost structure

uncoils, the behavior is different. The results presented lead to the conclusion that a

proportionality relationship exists between the depth to width ratio and the length of

maximum amplification 2π/β, for the critical eigenmodes, so as L/D, and consequently

LP /D increases, the length of the perturbation in the spanwise direction tends to increase

in similar proportion. These laws of behavior are valid as long as the morphological shape

of the eigenmodes is similar, once the variation of L/D produces significant differences the

expressions are no longer valid.

Another observation is the apparent relation between the frequency of the eigenmodes,

expressed as the dimensionless Strouhal number, St, and the cavity aspect ratio L/D. For

a given aspect ratio there is little variation of the Strouhal number of the modes with

the Reynolds number or the incoming boundary-layer thickness, so each mode appears to

be associated with a characteristic frequency. Figure 4.32 is a plot of the dimensionless

frequency of the different eigenmodes versus the aspect ratio. The frequency of some

stable eigenmodes is also considered. Using the results of the present analysis a correlation

between the two parameters can be constructed, which follows a potential law described

in equation 4.5.

St = K1(L/D)K2 (4.5)

The parameters K1 and K2 for the different modes can be found in table 4.8 (right).

Very few results can be found in the literature on these three dimensional structures with

low Strouhal. Nevertheless, the recent work of Basley et al. [16] presents an experimental

study on the open cavity flow in which a broad band of frequencies corresponding with

three dimensional structures can be found. The range of those frequencies match those

found here, as do the results presented in Bres and Colonius [20].


Table 4.8: The parameters C1 and C2 of equation (4.4) for the different modes(left) and the

parameters K1 and K2 of equation (4.5) for the different modes (right).

Mode C1 C2 Mode K1 K2

Mode I low β branch 112.1 3.79 Mode II 0.0069 0.696

Mode I high β branch 376.0 2.35 Mode I 0.0167 0.551

Mode II low β branch 234.5 3.11 Mode III 0.0340 0.635

Mode II high β branch 411.7 2.11 Stable Mode 0.0545 0.609

Stable Mode 0.0854 0.645

Figure 4.32: Strouhal number for the main oscillating modes versus the length-to-depth aspect

ratio of the cavity, both in logarithmic scale. Mode I, Mode II and the third unstable mode are

in red circles, blue circles and empty circles respectively. The other two, in triangles, are stable

modes for the range of parameters studied. In black rhombi the dominant frequency of Bres and

Colonius [20], and the range of frequencies obtained by Basley et al. [16] on the crossed line.

76 4.4 Summary of the chapter

4.4 Summary of the chapter

The onset of three-dimensional instability of the incompressible flow over spanwise homo-

geneous flow in a rectangular cavity has been described. For a wide range of the parameters

involved in the problem, the features of the dominant perturbations have been studied, al-

lowing the extraction of relations between the different parameters and a description of the

neutral hypersurface, through cuts with several planes of the parameter space. Two main

three-dimensional modes appear in the majority of the cases studied: a bifurcated mode,

noted Mode II, with both a travelling and a stationary (and more unstable) lobe, as well

as a purely travelling mode, noted Mode I. The stability behaviour of those modes with

variations of all the parameters has been reported, allowing predictions of the stability of

any configuration sufficiently close to critical conditions using only geometrical features

of the cavity. In particular, the relations between the characteristic Strouhal number of

the different eigenmodes, their critical Reynolds number and the L/D ratio have been

characterised. Such information should be useful not only in order to predict instabilities

of a given open cavity, but also as basis for reduced-order-models of global flow instability

and subsequent flow control, both theoretical and experimental.

5The two-dimensional limit

Contents

5.1 Convergence of BiGlobal analysis on β → 0 . . . . . . . . . . . . 77

5.2 Shear layer modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.1 Characteristic frequency . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 On eigenmode amplification . . . . . . . . . . . . . . . . . . . . . 81

5.1 Convergence of BiGlobal analysis on β → 0

The two-dimensional analysis of the flow over a rectangular cavity has been widely studied

in the literature. For the sake of completeness, a two-dimensional BiGlobal (β → 0)

analysis of the stability of the laminar two-dimensional incompressible cavity flow is first

discussed.

The first concern is with the numerical stability of the code when β → 0. As it can

be seen in de Vicente [27] the problem becomes singular for β = 0, and relatively low β

lead to ill conditioned but solvable problems. For these cases the behaviour of the two

dimensional structures can be assimilated to the one of structures with high enough Lz.

In figure 5.1 the evolution of the growth rate of the two least stable eigenmodes on

the 2D limit is shown for a test case. It can be seen that in the range of β ∼ 0.01 to

β ∼ 10−6 the eigenvalues do not change too much, while for β smaller than that numerous

spurious eigenvalues start to appear, and the two eigenvalues jump to a new value. It is

unlikely that there is something in the physics of the problem that changes at Lz/D ∼ 106,

given that β ∼ 10−6 is the range in which the eigenvalues related to the ill conditioning

of the problem grow. Consequently it is safe to assume that the eigenvalues on β = 0.01

sufficiently describe the two dimensional instabilities. All the computations presented in

the following 2D section are made with said β = 0.01.

77

78 5.2 Shear layer modes

Figure 5.1: Evolution of the growth rate of the two least stable eigenvalues on the two dimensional

limit, in a range of β ∼ 10−2 to β ∼ 10−9 for L/D = 2, ReD = 1500 and the constant δ−1.

5.2 Shear layer modes

The relevance and behaviour of the shear layer modes has been explained in Chapter 1.

This kind of structures consist on a feedback loop between the shear layer disturbances

and the acoustic disturbances in the impinging jet. Small perturbations in the shear layer

force periodical movements up and down (Kelvin-Helmholtz instabilities), which in turn

excites acoustic noise to develop from the impinging edge, where vortex shedding can

occur. These acoustic disturbances, which in the incompressible regime can be consid-

ered as instantaneous, travel upstream and excite the shear layer oscillations, closing the

feedback loop. These modes, extensively documented in Gloerfelt [41], follow Rossiter

semi-empirical formula,

Stn =n− γ

M+ 1κ

, n = 1, 2, 3, ..., (5.1)


where M is the Mach number, and γ and κ are empirical constants, given that the

characteristic length of the cavity is related to the appropriate resonance frequency. In

figure 5.2 the velocity profiles of the reconstruction of the total flow consisting of the base

flow plus the first of those shear layer modes is shown for a L/D = 2, and ReD = 2400 test

case. The shedding of vortices due to the impingement of the shear layer on the trailing

edge can be seen in the figure, as well as the resulting perturbation of the recirculating

vortex.

Figure 5.2: Velocity profiles of the reconstructed 2D flow with the first shear layer mode. Stream-

wise velocity on the top, and wall normal velocity on the bottom.


5.2.1 Characteristic frequency

In figure 5.3 the characteristic frequency of the least stable 2D eigenmodes in the range

of ReD and δ0/D studied (shown in table 4.5) is presented for L/D = 2 and L/D = 3.

0

0.02

0.04

0.06

0.08

0.4 0.5 0.6 0.7 0.8 0.9 1StL

0 /D

Figure 5.3: Variation of the dimensionless frequency (Strouhal based on cavity length StL) of

the two least stable eigenmodes (circles ◦ and rhombus � respectively) in the 2D limit with the

incoming boundary-layer thickness. Results for L/D = 2 in empty symbols (◦), and L/D = 3 in

full symbols(•). In grey shades, the range of Strouhal numbers obtained by Sarohia [90], and in

red symbols single points of several L/D = 2 works. The 2M2 run of Rowley et al. [87] in squares

(�), the Bres [19] M= 0.3 run as a rhombus ( ), the lower runs of Yamouni et al. [112] in triangles

(�). Points from Basley et al. [16] as circles (©).

As it can be seen, the frequency of the two shear layer modes found in this analysis

matches the range predicted in the literature (Basley et al. [16], Bres [19], Rowley et al.

[87], Sarohia [90], Yamouni et al. [112]), as well as the values of the three-dimensional DNS

computations made, as it was explained in section 4.2. The reduction in the Strouhal num-

ber of the modes with the increase of the incoming boundary-layer thickness reported here

was already observed in Yamouni et al. [112]. Given the relevance of L in the behavior

of this eigenmodes it is more usual in the literature to show the results of the two dimen-

sional shear layer modes as a function of the length of the cavity divided by the incoming

boundary layer thickness, L/θ0. In figure 5.4 the same data can be seen in that form, for


ease of comparison with other works. Note that using this dimensionalization does not

lead to a total collapse of the data of all L/D.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

40 50 60 70 80 90 100 110

StL

L/ 0

Figure 5.4: Variation of the dimensionless frequency (Strouhal based on cavity length StL) of

the two least stable eigenmodes (circles ◦ and rhombus � respectively) in the 2D limit with the

length of the cavity dimensionalized with the incoming boundary-layer thickness, L/θ0. Results

for L/D = 2 in empty symbols (◦), and L/D = 3 in full symbols(•). In in red symbols single

points of several works: the 2M2 run of Rowley et al. [87] in squares (�), the Bres [19] M= 0.3 run

as a rhombus ( ), the lower runs of Yamouni et al. [112] in triangles (�) and points from Basley

et al. [16] as circles (©).

5.2.2 On eigenmode amplification

Regarding the amplification of those modes, the behaviour is consistent with previous

results in the literature: when the shear layer becomes thicker the Kelvin-Helmholtz in-

stability dampens until the layer becomes stable, as has been observed experimentally

(Gharib and Roshko [40], Rockwell and Knisely [83]). Nevertheless, an issue arises when

considering the stability of these shear layer modes. Since the structure that can be seen

in figure 5.2 is convected downstream, it is partially responsible for the global stability of

the mode, which is dependant on the resolution of the mode wake.


Figure 5.5: Streamwise velocity profiles of the first shear layer mode, for ReD = 2400 and

θ−1/D = 0.0337. Different length domains, from top to bottom xout = 8, 13, 19 and 21 respectively.


These modes are spatially unstable, that is, after the cavity there is a growing wake,

corresponding with the shedding of vortices from the cavity, and then that wake sometimes

dissipate, depending on the Reynolds number. The aforementioned spacial instability is

reflected on the temporal instability analysis through a special sensibility to the domain

and the boundary conditions. Our studies show that the variation of the frequencies of

the mode is small when the length of the domain is changed (less than 5%). However, the

growth rate varies greatly, to the point of affecting the stable or unstable nature of the

mode. Figure 5.5 shows the streamwise velocity of the leading eigenmode with varying

domain length, and figure 5.6 shows the variation of the amplification of the eigenmodes

as the domain length changes from xout = 8 to xout = 21.

Figure 5.6: Variation of the dimensionless frequency (Strouhal based on cavity length StL) and

amplification σ of the two least stable eigenmodes with the change of domain length xout.


As it can be seen in figure 5.6, the Strouhal of the two leading two-dimensional eigen-

values suffers little variation as the domain changes, while the amplification changes dras-

tically. The behavior with the change of domain, and the change of flow parameters is

consistent. If the mode is sufficiently resolved (i.e., the domain is not excessively short),

for a given domain length increasing the incoming boundary layer thickness makes the

mode more unstable, as expected, and so on, independently of the length of the domain.

Nevertheless, if the domain is kept constant through the parametric sweep the behavior

of the modes is consistent, and a neutral curve, similar to the one made for the 3D modes

can be made, but once the domain changes so does the growth rate, rending those neutral

curves moot. In any case, the general effect of the parameters on the stability behavior

can be seen on that neutral curves, and as such are represented in figure 5.7.

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1100 1600 2100 2600ReD

0 /D

Figure 5.7: Neutral curves in ReD vs θ0 for the two least stable two-dimensional modes, in circles

◦ and rhombus � respectively. Three-dimensional Mode I and Mode II in red and blue, as in

figure 4.17. Domain length kept constant at xout = 8.

The two dimensional stability results presented in that figure corresponding with runs

in which the computational domain was kept constant in extension, number of points

and boundary conditions (and the last domain goes up to xout = 8). It can be seen

that the behavior here is not linear, with a lobe of instability appearing for the second

Rossiter mode on a certain range of incoming boundary layer thickness. The first mode

seems to behave similarly, but it was impossible to map the shape of the lobe due to it

being sufficiently into the unstable zone of the second mode to make impossible to get


stable 2D DNS computations. Additional experiments have been made, using several

2D DNS codes, and the limit in which those shear layer modes start to grow has been

found to vary significantly with the computational domain, matching the variation on the

obtained eigenvalues. This sensitivity on the conditions explains why it was possible to

make predictions on the behavior of 2D modes using a tool that requires a 2D stationary

base flow. As explained in Section 3.2 the DNS used to compute the base flow employs an

artificial compressibility method, with certain boundary conditions, that allow the flow to

remain 2D stable longer than the prediction made in this case.

6Experimental campaign

Contents

6.1 The experimental campaign . . . . . . . . . . . . . . . . . . . . . 87

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2.1 Validity of eigenmodes in the saturated regime . . . . . . . . . . 89

6.2.2 Effects of three-dimensional boundary conditions . . . . . . . . . 93

6.2.3 On the symmetry breakings . . . . . . . . . . . . . . . . . . . . . 95

6.3 Concluding remarks on the experimental campaign . . . . . . . 95

6.1 The experimental campaign

The experiments were conducted in a recirculating water tunnel at the Laboratory for Tur-

bulence Research for Aerospace & Combustion (LTRAC). All data collection and analysis

on the experimental campaign was made by Dr. Jeremy Basley, in collaboration with the

other authors of the resulting publication, de Vicente et al. [31], mainly Dr. Javier de

Vicente and Dr. Julio Soria, as well as the author of this text.

Regarding the water tunnel, the test section is 5 m long of cross-section 500×500 mm2

and turbulence intensity in the core region is less than 0.35 %, as shown in Parker et al. [74].

The reader may refer to Basley [14] for details on the experimental campaign. This chapter

uses material extracted from de Vicente et al. [31], published in JFM earlier this year. The

focus is here on the results and analyses directly related to the work of the author. The

reader may refer to the original and complete article for an exhaustive description, and for

the main results on the saturated flow. So, In what follows the main results related with

the relation between the linear analysis and the experimental campaign are reproduced.

The experimental set-up is sketched in Figure 6.1. The test plate is mounted vertically

in the middle of the test section. The 50 mm deep, D, 100 mm long, L, cavity spans

the water tunnel and is located 6.34 D from the leading edge of the plate. The distance

from flat plate water-tunnel walls is nominally F = 225 mm, such that F/D = 4.5. The

87

88 6.1 The experimental campaign

θ0 �x

�y

�U

D

L

Figure 6.1: Sketch of the experimental set-up. Dimensions are given in millimetres. The laser

sheet (at y = −0.1D) is represented in a close-up on the L = 2D – shaped cavity. High resolution

images require three cameras to span the cavity.

results presented in the following have been obtained for two mean free-stream velocities

UA = 29.5±0.8 mm/s and UB = 47.0±0.9 mm/s, which correspond to Reynolds numbers

ReD of 1500 and 2400, respectively.


6.2 Discussion

The BiGlobal stability analysis and velocity experimental measurements do not strictly

highlight the same state of the system. Firstly, linear stability analysis (section 2.2)

is concerned with the onset and the nature of the flow instabilities whereas the exper-

iments (section 6.1) deal with the final state of non-linearly saturated dynamics. Fur-

thermore, BiGlobal analysis involves an ideal noiseless two-dimensional basic flow with

periodic spanwise boundary conditions, as opposed to real conditions and confinement of

the experiments, which might change the stability properties of the flow. In addition,

measurements only give access to partial imperfect information, with uncertainties, being

also only a modelization of a real engineering problem to be compared with the complete

three-dimensional structure of the eigenmodes of the flow.

However, from merging those different objects comes a further understanding of the

mechanisms governing the evolution of the system, from the theoretical onset of centrifu-

gal instabilities to the real flow.

6.2.1 Validity of eigenmodes in the saturated regime

The forthcoming discussion mainly relies on figure 6.2, which presents a side-by-side de-

scription of case B (ReD = 2400, θ0 = 0.0340) from both points of view: linear stability

analysis and experimental measurements. Nominal case B presents a greater variety of

linearly unstable modes (figure 4.14) than case A, for which the control parameters are less

critical (ReD = 1500, θ0 = 0.0432). Indeed, case B reveals richer dynamics since it leads

to a flow being a combination of a greater number of different structures. As a result,

the features discussed hereinafter, regarding the intrinsic instabilities in the permanent

regime, apply similarly to case A in a simpler manner.

In the entire figure 6.2 squared letters and dotted annotations in red refer to results

predicted using BiGlobal analysis while circled blue letters and shaded regions symbolise

results from the experimental campaign. Series of dots in 6.2(top) denote the branches of

eigenvalues corresponding to growing disturbances, represented in the β-St plane. Four

of these eigenvalues are identified with letters (A, B, C and D) inside a square. Letter

D corresponds to β = 11.8, the wavenumber of maximum amplification for branch I,

90 6.2 Discussion

while letter A to β = 6.3, the wavenumber associated with the second local maxima of

amplification of the same branch (see figure 4.14 and figure 4.15 for more details).

Letter C points to two different values of β in the most unstable disturbance, Mode II:

β = 12 corresponding to maximum amplification in the stationary branch (II.a) and β =

8.5 just before the bifurcation in the oscillatory branch of the same Mode (II.b). Finally,

letter B indicates another mode of the same branch II.b, but with different properties

(β = 7.8, St = 0.0054).

In the same sub-figure, shades are a qualitative representation of the energy in this

β-St plane according to the space-time Fourier analysis of the experimental data. Dashed

lines, labelled by circled letters A to D, denote four characteristic frequencies of the Fourier

spectrum.

In the bottom part of figure 6.2, the spatial modes associated with the four items A to

D are depicted through their streamwise and spanwise velocity fields. The Fourier modes

(left column) and BiGlobal eigenmodes (right column) are discussed side-by-side. To that

end, velocity profiles have been extracted from a three-dimensional reconstruction of the

eigenmodes in the plane y = −0.1 D (as in the experiments). For the sake of clarity each

BiGlobal mode is only depicted in the area of most resemblance with the experimental

Fourier mode.

In order of increasing Strouhal number, the first modes to be considered are those cor-

responding to steady disturbances (denoted by letter C in figure 6.2). According to linear

analysis there is a range of wavenumbers (9 � β � 19) for which the stationary branch

of Mode II is unstable. In the experiments, wavenumbers for steady structures match the

BiGlobal predictions, but with a tendency to concentrate nearby the endwalls of the rig.

Within the uncertainty of experiments and in real conditions, ”quasi-steady” dynamics

can also be considered with regards to stationary eigenmodes. For instance, the structures

present on the right side of the Fourier mode (C) in figure 6.2(bottom) resemble those pre-

dicted by the linear stability analysis in the stationary branch II.a. The most coherent

and energetic structures correspond to β 12, which is the wavenumber for maximum

amplification according to BiGlobal analysis. Other coherent structures visible near the

left wall in the experiments for the same Fourier mode (C) exhibit a tilted shape, charac-

teristic of travelling waves. These slow-moving structures likely belong to the oscillatory

branch of the same mode (II.b) close to the bifurcation. Indeed, streamwise and spanwise

velocity components of the linearly reconstructed flow present a qualitative morphological

similarity with the experimental data in the region of the cavity where the mode appears.


Figure 6.2: Main flow features at ReD = 2400 for both experimental and linear stability analysis. (Top)

BiGlobal unstable eigenvalues (dots) and qualitative schematic depiction of the most energetic modes in the

experiments (shades) in the β-St plane. (Bottom) Velocity fields related to the four representative modes

highlighted in the upper figure. Left column corresponds to global Fourier modes from the experimental

dataset. Right column presents the reconstructed flow using BiGlobal analysis. For each mode, streamwise

velocity (top) and spanwise velocity (bottom) are shown.

92 6.2 Discussion

However, the coherent structures in the experiments are not dominant in terms of energy,

whereas the corresponding BiGlobal eigenmodes are associated with the largest growth

rates. It is important to point out that the mode with the highest growth rate in linear

analysis does not necessarily have to be the most energetic mode in the saturated regime.

The oscillatory branch II.b remains unstable as the Strouhal number increases, while the

associated spatial wavenumber decreases. This disturbance seems to correspond to a con-

tinuum of modes observed in the experiments. For instance, the dynamics depicted in the

Fourier mode (B) exhibit energetic coherent structures consistent with the eigenmodes (B)

existing in the same range of space-time scales, except for a slight reduction of the Strouhal

numbers. As expected, the velocity fields are morphologically similar to those previously

observed in modes (C), given that it is the same branch with a different wavenumber.

From the BiGlobal analysis, the most linearly unstable oscillatory branch is Mode I, for

both case A (ReD = 1500) and case B (ReD = 2400), corresponding to Strouhal numbers

such that St 0.025. In case B, although the fastest-growing eigenmode is associated

with β = 11.8 and correspond to St = 0.027, a broad range of spanwise waves are actually

unstable (for 4 � β � 18). The reconstruction of the most unstable configuration of this

branch is shown on the right side of figure 6.2(bottom), denoted with letter D. From the

experiments, the frequency band St 0.027 is associated with broad-banded dynamics

involving wavenumbers in the range 5 ≤ |β| ≤ 15. The global Fourier mode (D) seen in

figure 6.2(bottom) exhibits many patterns recalling the spatial structure of various eigen-

modes of branch I. This suggests that the saturated dynamics observed in the experiments

are composed of a continuum of waves deriving from intrinsic instabilities pertaining to

the unstable branch of Mode I.

Finally, the most prominent dynamics revealed by experimental datasets are associated

with Strouhal numbers such that 0.013 ≤St≤ 0.023. In both cases A and B, these domi-

nant features consist of highly coherent right or left travelling waves, corresponding to a

well defined wavelength λ D (|β| 2π). These structures have been encountered as

pairs of counter-propagating waves or as a stand-alone pulsating pattern when and where

two waves overlap. A global Fourier mode of such a travelling wave can be favourably

compared to an eigenmode from the branch I corresponding to the same β = 2π, as seen

with (A) in figure 6.2. In fact, the velocity fields are qualitatively analogous in a wide sec-

tion of the y = −0.1 D plane. However, the Strouhal numbers exhibited by the saturated

dynamics do not correspond to any linearly unstable eigenvalues, neither from branch I

(St 0.025) nor branch II.b (St < 0.010). Such a discrepancy between BiGlobal eigen-

modes and Fourier modes extracted from the real flow in the permanent regime is caused


either by nonlinearities or by the effects of solid boundary conditions on the stability

properties of the base-flow.

6.2.2 Effects of three-dimensional boundary conditions

As opposed to the two-dimensional base-flow around which the linear stability analysis

is performed, the real flow investigated experimentally involves solid boundary conditions

caused by endwalls located at z = ±5D. Such boundary conditions likely lead to the

creation of Bodewadt (Ekman-like) layers of opposite sign near both endwalls. The effect

of endwall layers on the centrifugal instabilities in cavity flows has been observed first

in lid-driven cavity flows (Albensoeder and Kuhlmann [3], Albensoeder et al. [5], Chiang

et al. [24], Koseff and Street [61, 62, 63], Shankar and Deshpande [94]). Endwall layers

are usually modelled as slow-rotating centripetal disks making the junction between the

main recirculation and rigid boundaries. Guermond et al. [48], Migeon et al. [68] notably

demonstrated that Bodewadt layers inject momentum through the centreline of the main

recirculation. In the case of confined flows such as lid-driven cavities, Bodewadt layers

would hence draw the outer edge of the inner-flow from the mid-span region towards the

endwalls. This would imply a spanwise drift of the centrifugal instability vortices, which

coil along the outer region of the main flow, and a consequent increase of the effective

Strouhal number, which is not seen here.

Similar dynamics have been observed by Faure et al. [35, 36] for open cavities of aspect

ratio around L/D ≤ 1.25. For such geometries, the most linearly unstable perturbation

over the two-dimensional base-flow is known to be a stationary disturbance (Bres and

Colonius [20], de Vicente [27], Meseguer-Garrido et al. [66], Pastur et al. [75]). On the

other hand, for larger L/D ratios the more complex geometry of the main recirculation

vortex results in increasing growth rates of oscillatory eigenmodes. Contrary to steady

modes, these unsteady modes derive into intrinsically travelling waves. Effects of endwalls

is hence more difficult to foresee.

A second effect regarding the effect of the walls was described by Shankar and Desh-

pande [94]. The authors observed the discrepancy between the 3-D and 2-D velocity

profiles for increasing Reynolds numbers due to the influence of endwall vortices. These

vortices not only provoke an increase in the spanwise flow but also slow down the main

centrifugal recirculation on the cavity, and that braking increases with Reynolds number.

So, the confinement causes the decrease in the velocity of the centrifugal perturbation,

forcing the stability properties of the base-flow to change. Indeed, Bres and Colonius [20]

94 6.2 Discussion

have asserted that, at the first order, the Strouhal numbers associated with oscillatory

eigenmodes are conditioned by the time required for a perturbation to travel along the

recirculation. From that, a deceleration due to Bodewadt layers could decrease the intrinsic

frequency of the spanwise waves coiling onto the recirculation, despite the influence of

the drift. This effect would become stronger for waves closer to the endwalls. Such an

hypothesis could explain the different Strouhal numbers between BiGlobal eigenmodes and

Fourier modes for the most energetic experimental mode (A) in figure 6.2. In figure 6.3 a

comparison between the streamwise velocity profiles in the experimental y/D = −0.1 plane

for the base flow for the linear analysis and the mean flow in the experiments. Both case

A and B are shown, taking into account the uncertainties of the measures. It can be seen

that, as expected, the experimental values of the velocity are smaller, suggesting a braking

in the main vortex. Also, this reduction of the velocity is greater in the higher Reynolds

case, which is consistent with the findings in Shankar and Deshpande [94]. This means

that the braking phenomenon is a plausible explanation to the reduced Strouhal numbers

reported in the experiments, although other possibilities are explored in Chapter 7.

0 0.5 1 1.5 2

−0.02

0

0.02

0.04

0.06

0.08

0.1

x/D

U

U0

2D Base−flow (BiGlobal analysis)3D mean−flow (experiments)

0 0.5 1 1.5 2

−0.02

0

0.02

0.04

0.06

0.08

0.1

x/D

U

U0

2D Base−flow (BiGlobal analysis)3D mean−flow (experiments)

Figure 6.3: Streamwise profiles of streamwise velocity U/U0 for case A (ReD = 1500) on the left

and case B (ReD = 2400) on the right. The profile obtained of the 2D base-flow used by BiGlobal

analysis is extracted from the range −0.12 � y/D � −0.09 (black), to represent the uncertainty on

the position and thickness of the laser-sheet. The profile issued of the 3D mean-flow, experimentally

measured in the zx-plane at y = −0.1D is extracted from the range −3 � z/D � 3 (blue), to take

into account spanwise variations.


6.2.3 On the symmetry breakings

The nonlinearly saturated flow exhibits asymmetries, which are, by definition, absent of

the (periodic) eigenmodes obtained through BiGlobal analysis. In particular, the question

of why the counter-propagating dominant waves are not symmetrical can arise.

Two plausible explications arise to justify this phenomenon. The symmetry breaking

may be caused by facility-dependent effects or to be inherent to the sensitivity of the flow

to initial conditions.

As always, experimental conditions are characterised within uncertainties. The sources

of possible experimental bias that could lead to spanwise asymmetries are: (a) an imperfect

cavity geometry, or a crooked velocity profile due to (b) water-tunnel design or (c) angular

discrepancy in cavity orientation.

Uncertainties (a) & (b) constitute systematic biases. They are ruled out since asymme-

try changes from one recording to another. On the other hand, (c) is concerned with the

alignment of the rig with z-axis (see de Vicente et al. [31] for a sketch of the setup). That

alignment could vary by about ±1 mm over the span S = 500 mm, corresponding to an

angular discrepancy of ±0.11°. Such an uncertainty of only ±0.1% cannot explain alone

the symmetry breaking. Consequently, an intrinsic sensitivity of the dynamics should

rather be considered.

In fact, the reconstructed flow obtained from stability analysis gives no prevalence to

left-travelling, right-travelling or pulsating structures as it was explained in section 4.1.1.

The smallest variation in the initial conditions hence causes the real flow to break sym-

metries by selecting a particular pattern.

6.3 Concluding remarks on the experimental campaign

The main goal of this section is to use both linear stability analysis and experiments to

cover the evolution of centrifugal instabilities in an open cavity flow from their onset to

their observation within the nonlinearly saturated state. The intrinsic stability properties

of the 2D base-flow were fully investigated through an extensive study of the parameter

space in Chapter 4 and the features changed by nonlinear effects or/and real boundary

conditions are identified and studied.

96 6.3 Concluding remarks on the experimental campaign

Overall, for the ReD = 2400 case the range of wavenumbers corresponding to growing

perturbations is as broad as 2.6 � β � 19, showing that this is a point in the parameter

space far from the stability threshold. The three-dimensional organisation associated with

each family of eigenmodes has been identified and characterised, thus allowing to determine

in advance some of the main agents involved in the real flow with low computational cost.

On the other hand, the experimental investigation of the real flow in the permanent

regime brings more insight into the dynamics that are actually selected by the real flow

beyond the linear transient growth, and once real boundary conditions are set, such as

a noisy incoming flow, lateral walls imposed to the cavity or even the influence of the

wake. Experimental measurements of centrifugal instabilities remain challenging in open

cavities, since those three-dimensional dynamics involve particularly low frequencies and

are greatly perturbed by the normally unstable shear layer above the cavity. Here, time-

resolved high-resolution PIV measurements were performed in a spanwise plane parallel

to the bottom of the cavity. Applying space-extended time-Fourier transform to such

experimental data allowed the identification of the coherent structures associated with

any given Strouhal number. The hypothesis of spanwise-waves has been confirmed by

experimental results, with dynamics in the range of unstable wavenumbers predicted by

linear stability analysis. Most of the eigenmodes were recovered within the real flow, in

spite of different lateral boundary conditions.

It must be noted that eigenmodes are recovered only locally, and that they can be

distorted. Indeed, the saturated dynamics are strongly modulated in amplitude, resulting

in local states, and those states change with time, that is, spanwise waves can also become

more or less dominant in terms of energy at different times. In BiGlobal analysis, travelling

eigenmodes have some degrees of freedom: the composition of the real and imaginary part

of the eigenmodes can form structures that travel right, left, or that pulsate without

shifting. Additional conditions in the real flow, confinement, noise in the upstream flow,

etc. take away that degree of freedom, and a concrete structure is formed. In the case of

several of those modes locally coexisting, they can overlap to produce interferences and

standing waves can appear.

In the saturated flow the fastest growing family of steady eigenmodes (II.a) is fairly

recovered as broad-banded spatial structures associated with the slowest dynamics (for

St→ 0). Similarly, the branch of Mode (I) corresponds to broad-banded dynamics at

0.025 � St � 0.03. On the other hand, spanwise waves associated with Strouhal num-

bers St� 0.01 exhibit narrower ranges of space-scales, which is consistent with eigen-

modes from branch (II.b). The most energetic travelling waves observed in the permanent


regime, though, partially depart from the linear stability results. Those highly coher-

ent waves strikingly resemble the eigenmodes from the unsteady low β branch of Mode

(I). However, they are associated with time-scales such that 0.013 � St � 0.023, lower

than the Strouhal numbers predicted for those structures by the linear stability analy-

sis. The hypothesis presented here to explain those different time-scales relies upon the

modification of the base-flow due to confinement effects. The presence of endwalls could

be responsible for a braking of the main recirculation, leading to slower travelling waves

in the three-dimensional base-flow, relatively to the two-dimensional base-flow, used by

BiGlobal stability analysis.

7Preliminary study on saturation

Contents

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.2 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.1 Regime I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.2 Regime II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.3 Regime III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.3.4 Regime IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.3.5 Regime V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 Concluding remarks on the study on saturation . . . . . . . . . 109

7.1 Introduction

Section 6.2.2 elaborated on the possibility that the apparent reduction of the characteristic

frequencies of the most energetic eigenmode from the theoretical value predicted by the

linear analysis is a consequence of the presence of the span-wise walls, which had the

effect of slowing down the main centrifugal recirculation of the cavity, thus reducing the

characteristic Strouhal number of these structures. Other possible explanations for this

phenomenon are the saturation of the flow, or the non-linear interaction between several

unstable eigenmodes.

A preliminary study was presented in Meseguer-Garrido et al. [65], trying to separate

these three effects. A three-dimensional DNS computation was performed for the same

flow parameters of cases A and B of de Vicente et al. [31], as presented in Chapter 6, but

with periodic boundary conditions. Thus, the effect of the end-walls was neglected, and the

restriction on the span-wise wavenumber β limited the number of eigenmodes interacting,

leaving the saturation as the main mechanism present in the study. The main finding of

99

100 7.2 Methodology

that paper is that the reduction of characteristic Strouhal number reported in de Vicente

et al. [31] occurs also in absence of span-wise walls. The work of Meseguer-Garrido et al.

[65] relied on the study of instantaneous flow-fields (snapshots) extracted from a three-

dimensional DNS computation, as well as on the evolution of the flow variables at one

point. A deeper study of the data is however required to fully understand the physics

behind it, as no conclusions could be reached once the flow became complex enough.

In the present chapter, the original DMD algorithm Schmid [92] is applied to the incom-

pressible fluid flow over a rectangular open cavity from the linear to the saturated regime,

as detailed in Meseguer-Garrido et al. [65], and in section 3.5, in order to understand

the evolution of span-wise instabilities of the flow and the interactions between different

dynamic modes. The numerical solutions required to construct the data-sequences of snap-

shots were obtained by means of a three-dimensional non-steady DNS solver, explained

in section 3.4. The most relevant DMD modes and associated oscillation frequencies are

then compared to the ones obtained using linear stability analysis, allowing us to assess

the accuracy of the aforementioned snapshot-based decomposition. The numerical details

of the application of the different methods used for the present investigation are explained

in section 7.2. The BiGlobal and DNS results, and the DMD analysis performed are pre-

sented in section 7.3. Finally, the most significant conclusions obtained are summarized

in section 7.4.

7.2 Methodology

7.2.1 Problem description

A schematic representation of the flow configuration was depicted in figure 4.3. The case

studied here corresponds with the experimental Case B, with ReD = 2400 and θ0/D =

0.036 in a cavity of L/D = 2, as explained before.

As it was explained in Chapters 4 and 6 in the range of parameters close to the limit

of stability, the linear analysis performed shows the presence of two main branches of

unstable eigenmodes, as can be seen in figure 7.1 (top). The mode that becomes unstable

at lower Reynolds number, Mode I, is a travelling disturbance which is more unstable in

the proximity of β 6 and β 12. Mode II, the second to become unstable, is stationary

at higher β, while undergoing a bifurcation at β 9, resulting in a pair of complex

conjugate eigenvalues for values of β lower than that. The range of unstable eigenvalues


of both Mode I and II in the St-β plane, for the chosen parameters in this study, can be

seen in figure 7.1 (bottom).

Figure 7.1: Neutral curves for the L/D = 2 cavity in the ReD vs β plane, and selected Case

B (top). StD vs β map of unstable eigenmodes for Case B, and selected β by the periodicity

conditions of the DNS computations (bottom).

102 7.2 Methodology

7.2.2 Computational Setup

The DNS code employed to compute the flow in the mentioned cases is detailed in sec-

tion 3.4. As it was previously mentioned, the span-wise length of the computational

domain was selected in order to reduce the amount of interaction between the different

modes. By using a Lz = 2π/6 only the modes of spanwise wavenumber, β, multiple of 6

can appear, and those correspond with the β maximum amplification of the linear modes

(β = 6 and β = 12), as can be seen in figure 7.1 (bottom). The amplification of the modes

on β = 18 is very small, so it is unlikely that they appear in the simulation. Also, span-

wise periodic boundary conditions were used to avoid the effect of the span-wise walls, as

it was previously explained.

The chosen flow conditions were the same as in the linear analysis, with the exception

of the Mach number, i.e. M= 0.1. The results obtained in the linear growth regime

with this low Mach number are identical to those obtained with the incompressible linear

analysis, as can be seen in Meseguer-Garrido et al. [65], and in section 4.2.2. A block

structured mesh with 104 hexahedral subdomains was used, each of those subdomains

with (nx, ny, nz) = 15. For the chosen conditions the two-dimensional stationary solution

was extended periodically in the whole domain, and then a random noise of 10−8 was

introduced to kick-start the growth of the linearly unstable modes. The flow variables in

a point in the middle of the cavity were recorded, and snapshots of the whole flowfield

were saved each 10 non-dimensional time units. The L∞-norm of the span-wise velocity

component of the perturbed flow is monitored as a function of non-dimensional time on

the control point to determine the different regimes of the flow. Since the value of said

velocity component is zero in the two dimensional base flow, the whole effect corresponds

to perturbation, allowing to see the modes in greater detail, without dealing with the base

flow.

In the investigated fluid flow over a rectangular open cavity, five instability regions of

span-wise velocity component are identified from the linear to the saturated regime, as

reported in Meseguer-Garrido et al. [65]. Those five regimes can be seen in figure 7.2.

A DMD (see section 3.5 for details on this tool) was thus applied to regions II to V.

Convergence for each regime was accepted when the residual norm of the DMD and the

eigenvalues corresponding to the most unstable dynamic modes became independent of

the number of snapshots comprising the matrix V N1 (see section 3.5).


Figure 7.2: Temporal evolution of the absolute value of the span-wise velocity component in the

control point.

7.3 Results

7.3.1 Regime I

The preliminary study conducted in Meseguer-Garrido et al. [65] shows that the behavior

of the DNS in the linear growth phase (region I of figure 7.2) matches perfectly with the

results predicted by the linear analysis, with the most unstable eigenmode appearing and

growing exponentially with the anticipated σ. Those results can be seen in section 4.2.2.

7.3.2 Regime II

After the linear growth phase there is a saturation of the leading mode (t 500), and

then the stationary mode starts to pulsate with a StD of 0.0098 in regime II. The physical

morphology of the mode does not change, and the oscillations gets dampened with time.

The Strouhal number of these oscillations is the same as the StD of the low β branch

of Mode II (StD = 0.0099), according to linear analysis. So the behavior seems to be

104 7.3 Results

Figure 7.3: DMD modes on regime II. On the top, situation in the StD vs β plane (left) of DMD

modes A and B (right). On the bottom, BiGlobal mode corresponding with point A (Mode II for

β = 12).

that after saturation the stationary mode starts vibrating. That vibration is progressively

dampened, and in this case, with strong selection of Lz the perturbations resonate at

the characteristic frequency of the same branch of said mode, in the low β regime, even

though the structures do not change their characteristic β = 12. This is a very interesting

behavior, as the mode tends to vibrate with a frequency which is somehow natural to

it (although for a different β), even though for this β it is inherently stationary. The


DMD analysis of regime II can be seen in figure 7.3. It shows a strong presence of a

stationary mode (A), with a pulsation of the same structure (B). The morphology of

these structures is qualitatively identical to those of the stationary branch of the Biglobal

analysis (figure 7.3 top right and bottom, respectively).

7.3.3 Regime III

After the oscillations characterizing regime II have almost completely dampened a new

mode starts to grow in regime III (t 1500 to t 2100). Figure 7.4 (top) shows two

instantaneous flowfields belonging to that period of time, compared with the linear com-

position of the low β branch of Mode I and the high β branch of Mode II of linear analysis

(bottom). As it was explained in section 4.1.1 the lack of normalization on the eigenvector

makes so the values of the constants in the composition gives no additional information, in

this case there is four degrees of freedom, corresponding to two pairs of α, and they were

chosen to make that composition as similar as possible to the DNS flow. The oscillating

mode has β = 6 and StD = 0.0194, by determining the frequency on figure 7.2. That

frequency is smaller than the predicted one from linear analysis (StD = 0.025), but the

identified mode is the same Mode I, as can be seen in figure 7.4. As it was previously

discussed, the reduction of the characteristic frequency of this dominant mode of β = 6

was already observed in the previous experimental work of de Vicente et al. [31]. These

observations point to the fact that the saturation process and the non-linear interaction of

the modes is enough to produce the St reduction, without the presence of spanwise walls.

Figure 7.5 shows the results obtained with DMD for the same regime III. In this case

the short period of time and limited sampling frequency available, coupled with the nature

of the growth of the second mode, makes the DMD convergence not as good as in other

cases. In any case, the two main structures that can be appreciated in this regime are the

aforementioned mode A, still stationary and strongly resembling the stationary branch

of BiGlobal Mode II, and a structure pulsating with StD = 0.019 (mode C in figure 7.5)

that bears some similarity with the linear structure of β = 6 of Mode I (shown in the left

part of figure 7.4 bottom). In this case the mere composition of linear modes is closer to

an instantaneous flowfield than the DMD modes, due to, as it was previously mentioned,

poor convergence motivated by the short length of this regime, and the varying nature of

mode C.

106 7.3 Results

Figure 7.4: Two instantaneous flowfields in region III (top). Composition of the two linear modes

that yields a similar flowfield (bottom).

7.3.4 Regime IV

The next step in the DNS evolution, regime IV (t 2100 to t 2900), features the

apparition of a secondary oscillation in counter-phase with the main one, producing a

shifting of the frequency to StD = 0.015. This regime is at first stable in amplification, until

t 2700. In this stage the structures of the isocontours of spanwise velocity component

no longer resemble easily constructable combinations of linear modes. This last frequency

is once more in the range of frequencies of high energy in the experiments of Chapter 6

and de Vicente et al. [31]. The results of the DMD analysis in this regime can be seen in

figure 7.6, and allow the separation of the features of the flow into identifiable structures.

Mode A still resembles the stationary branch of the linear mode, although the mode has


Figure 7.5: DMD modes on regime III. Situation in the StD vs β plane of DMD modes A and C.

Figure 7.6: DMD modes on regime IV. Situation in the StD vs β plane of DMD and modes A,

C, E and D.

varied morphologically. Since the structures coiling around the main recirculating vortex

have started breaking it, the modes are no longer restricted to coil around the exterior

108 7.3 Results

part of the vortex, but fill a greater fraction of the cavity space. A similar phenomenon

occurs to mode C, which still corresponds, as before, with the linear mode in figure 7.4,

but this time oscillating with a frequency of StD = 0.015 (also within the range of StD

of the dominant mode in de Vicente et al. [31]). Other structures appear here, one of

StD = 0.030, and β = 12, which corresponds with the aforementioned counter-phase

oscillation (mode E), and another one in which two different β can be seen (mode D), and

with a frequency in line with the linear mode, StD = 0.025. Since DMD does a separation

by frequency, not by spatial structure, the two structures which oscillate with StD = 0.025,

one with β = 6 and the other with β = 12, get combined into a single DMD mode. It

is reasonable to assume that the linear composition of different structures of the linear

Mode I of different β could produce structures similar to mode D. On the other hand those

two modes could correspond with structures related with the secondary instability of the

saturated flow, fruit of non-linear interaction between the different structures. It seems

relevant that the appearance of a mode of StD = 0.030, which does not appear in the

linear analysis, nor it was present in the experiments conducted in de Vicente et al. [31],

coincides with the reduction of the characteristic frequency of mode C from StD = 0.019

to StD = 0.015, which is exactly half of the one of mode E. There seems to be some

sort of coupling between those modes in frequency, with stable amplitude. That constant

amplitude changes after t 2700, leading to regime V.

7.3.5 Regime V

After the breaking of the amplitude-stable counter-phase oscillation on t 2700 another

change occurs, as the structures trespass the x−y plane in the middle of the computational

domain. This produces a drastic increase in the complexity of the structures present in

the flow, as different packs of waves start travelling left or right, instead of just staying

stationary, or pulsating without changing position. These complex structures character-

ize the behavior of regime V (t 2900 onwards). The possibility of having travelling

structures appears in the linear analysis, as the not travelling ones are just a single com-

bination of the complex conjugate eigenvectors, but once the flow starts developing this

kind of structures, those shift in short periods of time. This might explain the not so good

convergence of the DMD technique for the present regime. The physics of the problem still

select some natural frequencies, so the DMD modes are extracted with constant Strouhal

numbers, but the shape of the modes varies greatly. Nevertheless, the results of the DMD

of regime V can be seen in figure 7.7, and some of the familiar structures are still present,

like the stationary mode A, the dominant mode C, or mode E. The nature of the mode on


Figure 7.7: DMD modes on regime V. Situation in the StD vs β plane of DMD and modes A, C,

E and F.

the StD = 0.021 to StD = 0.025 band experiences the most salient changes, as structures

of all the possible β (0, 6 and 12) merge in a single mode. An example of such is mode F

in figure 7.7.

7.4 Concluding remarks on the study on saturation

This work shows the power of the DMD tool to analyze a complex problem. The separation

of structures provided by this technique allows to identify the two most relevant linear

modes well beyond the linear regime of the DNS computation. The description of the

dominant structures in the five different flow regimes allows to better understand how the

process of saturation affects the morphology of the modes, as well as their characteristic

frequencies, without having to deal with the effect of the spanwise walls, unavoidable in an

experimental setup. The reported reduction of the characteristic Strouhal number of the

dominant mode from the value given by the linear analysis is found again here, indicating

that the effect is either a byproduct of the change of the secondary instability, or an effect

of the non-linear interaction of the different modes.

8Summary

and Future Directions

In the present thesis the BiGlobal instability analysis tool has been used to characterize

the onset of instability in the incompressible open cavity flow. Both two-dimensional and

spanwise periodic three-dimensional have been identified and thoroughly described. For

a wide range of the parameters of the problem the features of the leading perturbations

have been studied, allowing to construct expressions that relate the different parameters

and characteristics of the flow, as well as describe the neutral hypersurface through cuts

with several planes on the paremeter space.

Two main three-dimensional modes appear in the majority of the cases studied: a bi-

furcated mode, noted Mode II, with both a travelling and a stationary (and more unstable)

lobe, as well as a purely travelling mode (Mode I). The stability behaviour of those modes

with variations of all the parameters has been reported, allowing to predict the stability

of any configuration sufficiently close to critical conditions using only geometrical features

of the cavity. In particular, the relations between the characteristic Strouhal number of

the different eigenmodes, their critical Reynolds number and the L/D ratio have been

characterised. Such information should useful not only in order to predict instabilities of

a given open cavity, but also as basis for reduced-order-models of global flow instability

and subsequent flow control, both theoretical and experimental.

Also, the use of experiments has allowed to determine the validity BiGlobal tool, linear

in nature, well outside the linear regime, and the features changed by nonlinear effects

or/and real boundary conditions were identified and studied. The study of the dynamics

on the saturated regime was continued through a preliminary study on saturation through

numerical tools, using a three-dimensional DNS to compute a simple periodic case, and

using DMD to extract the temporal behaviour of the flow during saturation. The reported

reduction of the characteristic Strouhal number of the dominant mode from the value given

by the linear analysis is found again here, indicating that the effect is either a byproduct

of the change of the secondary instability, or an effect of the non-linear interaction of the

different modes.

112

Regarding future directions of this work, the study of the dynamics of saturation is

going to be followed, using three-dimensional DNS simulation that are wall bounded in

the spanwise direction, and also periodic with different lengths. Using those different

conditions the true nature of the decrease in Strouhal number should be determined. The

identification of the secondary instabilities is also being studied, first through the use of a

BiGlobal code that allows for a base flow with the three velocity components, and using

an average flow of the three-dimensional saturated flow, and in future works, through the

use of a TriGlobal tool, or at least a BiGlobal with a fourier transformation in the base

flow. Lastly, efforts are being made in regard to the two-dimensional instabilities, and the

sensibility to the domain and boundary conditions.

Bibliography

[1] N. Abdessemed. Stability analysis of flow past a low-pressure turbine blade. PhD

Thesis. Imperial College, 2007.

[2] N. Abdessemed, S. J. Sherwin, and V. Theofilis. Linear instability analysis of low-

pressure turbine flows. J. Fluid Mech., 628:57–83, 2009.

[3] S. Albensoeder and H. C. Kuhlmann. Nonlinear three-dimensional flow in the lid-

driven square cavity. J. Fluid Mech., 569:465–480, 2006.

[4] S. Albensoeder, H. C. Kuhlmann, and H. J. Rath. Three-dimensional centrifugal-

flow instabilities in the lid-driven-cavity problem. Physics of Fluids (1994-present),

13(1):121–135, 2001.

[5] S. Albensoeder, H. C. Kuhlmann, and H. J. Rath. Multiplicity of steady two-

dimensional flows in twosided lid-driven cavities. Theor. Comp. Fluid Dynamics, 14:

223–241, 2001.

[6] F. Alizard, J. C. Robinet, and X. Gloerfelt. A domain decomposition matrix-free

method for global linear stability. Comput. and Fluids, 66:63–84, 2012.

[7] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontin-

uous Galerkin methods for elliptic problems. SIAM Journal of Numerical Analysis,

39(5):1749–1779, 2001.

[8] S. Bagheri. Koopman-mode decomposition of the cylinder wake. Journal of Fluid

Mechanics, 726:596–623, 2013.

[9] S. Bagheri, E. Akervik, L. Brandt, and D. S. Henningson. Matrix-free methods for

the stability and control of boundary layers. AIAA J., 47(5):1057–1068, 2009.

[10] A. Barbagallo, D. Sipp, and P. J. Schmid. Closed-loop control of an open cavity flow

using reduced-order models. Journal of Fluid Mechanics, 641:1–50, 11 2009. ISSN

1469-7645.

113

114 BIBLIOGRAPHY

[11] D. Barkley. Linear analysis of the cylinder wake mean flow. Europhysics Letters, 75

(5):750–756, 2006.

[12] D. Barkley and R. D. Henderson. Three-dimensional floquet stability analysis of the

wake of a circular cylinder. J. Fluid Mech., 322:215–241, 1996.

[13] D. Barkley, H. M. Blackburn, and S. J. Sherwin. Direct optimal growth analysis for

timesteppers. Int. J. Numer. Meth. Fluids, 57(9):1435–1458, July 2008.

[14] J. Basley. An Experimental Investigation on Waves and Coherent Structures in a

Three-Dimensional Open Cavity Flow. PhD thesis, Universite Paris-Sud – Monash

University, 2012.

[15] J. Basley, L. R. Pastur, F. Lusseyran, T. M. Faure, and N. Delprat. Experimental

investigation of global structures in an incompressible cavity flow using time-resolved

PIV. Exp. Fluids, 50(4):905–918, April 2011. DOI: 10.1007/s00348-010-0942-9.

[16] J. Basley, L. R. Pastur, N. Delprat, and F. Lusseyran. Space-time aspects of a

three-dimensional multi-modulated open cavity flow. Physics of Fluids, 25(6), 2013.

doi: 10.1063/1.4811692.

[17] J. Basley, L. R. Pastur, F. Lusseyran, J. Soria, and N. Delprat. On the modu-

lating effect of three-dimensional instabilities in open cavity flows. J. Fluid Mech,

Submitted and under revision, 2014.

[18] F. P. Bertolotti, Th. Herbert, and P. R. Spalart. Linear and nonlinear stability of

the Blasius boundary layer. J. Fluid Mech., 242:441–474, 1992.

[19] G. A. Bres. Numerical simulations of three-dimensional instabilities in cavity flows.

PhD thesis, California Institute of Technology, 2007.

[20] G. A. Bres and T. Colonius. Three-dimensional instabilities in compressible flow

over open cavities. J. Fluid Mech., 599:309–339, 2008.

[21] K. Butler and B. F. Farrell. Three-dimensional optimal perturbations in viscous

shear flow. Phys. Fluids, 4(8), 1992.

[22] L. N. Cattafesta III, S. Garg, M. S. Kegerise, and G. S. Jones. Experiments on

compressible flow-induced cavity oscillations. AIAA J., page 2912, 1998.

[23] K. Chen, J. Tu, and C. Rowley. Variants of dynamic mode decomposition: boundary

condition, koopman, and fourier analyses. Journal of Nonlinear Science, 22(6):887–

915, 2011.

BIBLIOGRAPHY 115

[24] TP Chiang, WH Sheu, and RR Hwang. Effects of the reynolds number on the eddy

structure in a lid-driven cavity. Int. J. Numer. Methods Fluids, 26:557–579, 1998.

[25] T. Colonius, C. W. Rowley, and V. Theofilis. Global instabilities and reduced-order

models of cavity flow oscillations. In V. Theofilis, T. Colonius, and A. Seifert, editors,

Proceedings of the AFOSR/EOARD/ERCOFTAC SIG-33: Global Flow Instability

and Control Symposium - I, Sept. 22-26, 2001, Crete, Greece, Sept. 2001. ISBN-13:

978-84-692-6244-3. Fundacion General UPM. ISBN 84-89925-65-8.

[26] J. D. Crouch, A. Garbaruk, and D. Magidov. Predicting the onset of flow unsteadi-

ness based on global instability. J. Comput. Phys., 224:924–940, 2007.

[27] J. de Vicente. Spectral Multi-Domain Method for the Global Instability Analysis of

Complex Cavity Flows. PhD thesis, Universidad Politecnica de Madrid, 2010.

[28] J. de Vicente, E. Valero, L. M. Gonzalez, and V. Theofilis. Spectral multi-domain

methods for biglobal instability analysis of complex flows over open cavity configu-

rations. 34th Fluid Dynamics Conference and Exhibit, Portland, Oregon, June 28 –

July 1 2004. AIAA Paper 2004-2544.

[29] J. de Vicente, D. Rodriguez, V. Theofilis, and E. Valero. Stability analysis in

spanwise-periodic double-sided lid-driven cavity flows with complex cross-sectional

profiles. Comput. and Fluids, 43(1):143–153, 2010.

[30] J. de Vicente, P. Paredes, E. Valero, and V. Theofilis. Wave-like disturbances on the

downstream wall of an open cavity. Number 3754 in 6th AIAA Theoretical Fluid

Mechanics Conference, Honolulu, Hawaii, June 27-30 2011. AIAA.

[31] J. de Vicente, J. Basley, F. Meseguer-Garrido, J. Soria, and V. Theofilis. Three-

dimensional instabilities over a rectangular open cavity: from linear analysis to

experimentation. J. Fluid Mech., 748:189–220, 2014.

[32] P. G. Drazin and W. H. Reid. Hydrodynamic Stability. Cambridge University Press,

1981.

[33] D. Duke, J. Soria, and D. Honnery. An error analysis of the dynamic mode decompo-

sition. Experiments in Fluids, 52(2):529–542, 2012. doi: 10.1007/s00348-011-1235-7.

[34] B. F. Farrell and P. J. Ioannou. Generalized stability theory. Part I: Autonomous

operators. J. Atmos. Sci., 53(14):2025–2040, 1996.

116 BIBLIOGRAPHY

[35] T. M. Faure, P. Adrianos, F. Lusseyran, and L. R. Pastur. Visualizations of the flow

inside an open cavity at medium range reynolds numbers. Experiments in Fluids,

42(2):169–184, 2007.

[36] T. M. Faure, L. R. Pastur, F. Lusseyran, Y. Fraigneau, and D. Bisch. Three-

dimensional centrifugal instabilities development inside a parallelepipedic open cav-

ity of various shape. Experiments in Fluids, 47(3):395–410, 2009.

[37] E. Ferrer, J. DeVicente, and E. Valero. Low cost 3d global instability analysis and

flow sensitivity based on dynamic mode decomposition and high order numerical

tools. International Journal for Numerical Methods in Fluids, 2014.

[38] N. Fietier and M. O. Deville. Time-dependent algorithms for the simulation of vis-

coelastic flows with spectral element methods: applications and stability. J. Comput.

Phys., 186 (1):93–121, 2003.

[39] M. Gaster. A note on the relation between temporally increasing and spatially

increasing disturbances in hydrodynamic instability. J. Fluid Mech., 14:222–224,

1962.

[40] M. Gharib and A. Roshko. The effect of flow oscillations on cavity drag. J. Fluid

Mech., 177:501–530, 1987.

[41] X. Gloerfelt. Aerodynamic noise from wall-bounded flows: Cavity noise, chapter 0.

VKI Lectures, 2009.

[42] F. Gomez, S. L. Clainche, P. Paredes, M. Hermanns, and V. Theofilis. Four decades

of studying global linear instability: Progress and challenges. AIAA Journal, 50(12):

2731–2743, December 2012. doi: 10.2514/1.J051527.

[43] F. Gomez, R. Gomez, and V. Theofilis. On three-dimensional global linear instability

analysis of flows with standard aerodynamics codes. Aerosp. Sci. Techn., 32(1):223–

234, 2014.

[44] L. Gonzalez, V. Theofilis, and R. Gomez-Blanco. Finite element methods for viscous

incompressible biGlobal instability analysis on unstructured meshes. AIAA J., 45

(4):840–854, 2007.

[45] L. Gonzalez, V. Theofilis, and S. J. Sherwin. High-order methods for the numerical

solution of the BiGlobal linear stability eigenvalue problem in complex geometries.

Int. J. Numer. Meth. Fluids, (to appear):DOI:10.1002/fld.2220, 2010.

BIBLIOGRAPHY 117

[46] L. M. Gonzalez, M. Ahmed, J. Kuhnen, H. C. Kuhlmann, and V. Theofilis. Three-

dimensional flow instability in a lid-driven isosceles triangular cavity. Journal of

Fluid Mechanics, 675:369–396, 5 2011. ISSN 1469-7645.

[47] P. M. Gresho and R. L. Sani. On pressure boundary conditions for the incompressible

navier-stokes equations. International Journal for Numerical Methods in Fluids, 7

(10):1111–1145, 1987. ISSN 1097-0363.

[48] J.-L. Guermond, C. Migeon, G. Pineau, and L. Quartapelle. Start-up flows in a

three-dimensional rectangular driven cavity of aspect ratio 1:1:2 at re=1000. J.

Fluid Mech., 450:169–199, 2002.

[49] R. D. Henderson and D. Barkley. Secondary instability of the wake of a circular

cylinder. Phys. Fluids, 8(6):65–112, 1996.

[50] T. Herbert. Secondary instability of boundary layers. Annu. Rev. Fluid Mech., 20:

487–526, 1988.

[51] T. Herbert. Parabolized stability equations. Annual Review of Fluid Mechanics, 29:

245 – 283, 1997.

[52] C. P. Jackson. A finite-element study of the onset of vortex shedding in flow past

variously shaped bodies. J. Fluid Mech., 182:23–45, 1987.

[53] G. B. Jacobs, D. Kopriva, and F. Mashayek. Validation study of a multidomain

spectral code for simulation of turbulent flows. AIAA journal, 43(6):1256–1264,

2005.

[54] E. Janke and P. Balakumar. On the secondary instability of three-dimensional

boundary layers. Theor. Comp. Fluid Dynamics, 14:167–194, 2000.

[55] M. Jovanovic, P. Schmid, and J. Nichols. Sparsity-promoting dynamic mode decom-

position. Physics of Fluids, 26(2):024103 (22 pages), 2014.

[56] G. Em Karniadakis and S. J. Sherwin. Spectral/hp element methods for CFD. OUP,

2005.

[57] M. A. Kegerise, E. F. Spina, S. Garg, and L. N. Cattafesta III. Mode-switching and

nonlinear effects in compressible flow over a cavity. Phys. Fluids, 16:678–687, 2004.

[58] M. H. Kobayashi, J. C. F. Pereira, and J. M. M. Sousa. Comparison of several

open boundary numerical treatments for laminar recirculating flows. Int. J. Numer.

Meth. Fluids, 16(5):403–419, 1993.

118 BIBLIOGRAPHY

[59] D. Kopriva. Implementing Spectral Methods for Partial Differential Equations: Al-

gorithms for Scientists and Engineers. Springer, 2009.

[60] D. A. Kopriva. A staggered-grid multidomain spectral method for the compressible

navier-stokes equations. J. Comput. Phys., 244:142–158, 1998.

[61] J. R. Koseff and R. L. Street. The lid-driven cavity flow: A synthesis of qualitative

and quantitative observations. Trans. ASME: J. Fluids Eng., 106:390–398, 1984.

[62] J. R. Koseff and R. L. Street. On endwall effects in a lid-driven cavity flow. Trans.

ASME: J. Fluids Eng., 106:385–389, 1984.

[63] J. R. Koseff and R. L. Street. Visualization studies of a shear driven three-

dimensional recirculating flow. Trans. ASME: J. Fluids Eng., 106:21–29, 1984.

[64] R. Lehoucq, D. Sorensen, and C. Young. ARPACK Users’ Guide: Solution of Large

Eigenvalue Problems with Implicitly Restarted Arnoldi Method. SIAM Philadelphia,

1998.

[65] F. Meseguer-Garrido, J. de Vicente, and E. Valero. Three-dimensional analysis of

incompressible flow over an open cavity using direct numerical simulation: from

linear to saturated regime. 5980, Prato, Italy, June 5-8 2014. Instability and Control

of Massively Separated Flows.

[66] F. Meseguer-Garrido, J. de Vicente, E. Valero, and V. Theofilis. On linear instability

mechanisms in the incompressible open cavity flow. J. Fluid Mech., 752:219–236,

2014.

[67] I. Mezic. Analysis of fluid flows via spectral properties of the koopman operator.

Annual Review of Fluid Mechanics, 45:357–378, 2013.

[68] C. Migeon, G. Pineau, and A. Texier. Three-dimensionality development inside

standard parallelepipedic lid-driven cavities at re=1000. J. Fluids Struct., 17:717–

738, 2003.

[69] M. Morzynksi and F. Thiele. Numerical stability analysis of flow about a cylinder.

Z. Angew. Math. Mech., 71:T242–T248, 1991.

[70] T. Muld, G. Efraimsson, and D. Henningson. Mode decomposition on surface-

mounted cube. Flow Turbulence Combust, 88(3):279 – 310, 2012.

[71] M. D. Neary and K. D. Stephanoff. Shear-layer-driven transition in a rectangular

cavity. Phys. Fluids, 30(10):2936–2946, June 1987.

BIBLIOGRAPHY 119

[72] S. A. Orszag and M. Israeli. Numerical simulation of viscous incompressible flows.

Annual Review of Fluid Mechanics, 6:281–318, 1974.

[73] S. A. Orszag, M. Israeli, and M. O. Deville. Boundary conditions for incompressible

flows. Journal of Scientific Computing, 1(1):75–111, 1986.

[74] K Parker, K D von Ellenrieder, and J. Soria. Morphology of the forced oscillatory

flow past a finite-span wing at low Reynolds number. J. Fluid Mech., 571:327–357,

2007.

[75] L. R. Pastur, Y. Fraigneau, F. Lusseyran, and J Basley. From linear stability analysis

to three-dimensional organisation in an incompressible open cavity flow. Arxiv, July

2012.

[76] J. C. F. Pereira and J. J. M. M. Sousa. Influence of impingement edge geometry on

cavity flow oscillations. AIAA Journal, 1994.

[77] R. T. Pierrehumbert. A universal shortwave instability of two-dimensional eddies in

an inviscid fluid. Phys Rev Letters, 57:2157–2159, 1986.

[78] E. Piot, G. Casalis, F. Muller, and C. Bailly. Investigation of the pse approach for

subsonic and supersonic hot jets. detailed comparisons with les and linearized euler

equation results. Int. Journal of Aeroacoustics, 5(4):361–393, 2006.

[79] A. Powell. On edge tones and associated phenomena. Acustica, 3, 1953.

[80] F. Richecoeur, L. Hakim, A. Renaud, and L. Zimmer. Dmd algorithms for experi-

mental data processing in combustion. In Proceedings of the Summer Program 2012,

pages 459–468. Center for Turbulence Research, 2012.

[81] D. Rockwell. Prediction of oscillation frequencies for unstable flow past cavities.

Trans. ASME: J. Fluids Eng., pages 294–300, June 1977.

[82] D. Rockwell and C. Knisely. Observations of the threedimensional nature of unstable

flow past a cavity. Phys. Fluids, 23(425), 1980.

[83] D. Rockwell and C. Knisely. The organized nature of flow impingement upon a

corner. Journal of Fluid Mechanics, 93:413–432, 1987. ISSN 1469-7645.

[84] D. Rockwell and E. Naudascher. Self-sustained oscillations of impinging free shear

layers. Ann. Rev. Fluid Mech., 11:67–94, 1979.

120 BIBLIOGRAPHY

[85] D. Rodrıguez and V. Theofilis. Massively parallel numerical solution of the BiGlobal

linear instability eigenvalue problem using dense linear algebra. AIAA J., 47(10):

2449–2459, 2009.

[86] J.E. Rossiter. Wind-tunnel experiments on the flow over rectangular cavities at sub-

sonic and transonic speeds. Aeronautical Research Council Reports and Memoranda,

3438, October 1964.

[87] C. W. Rowley, T. Colonius, and A. J. Basu. On self-sustained oscillations in two-

dimensional compressible flow over rectangular cavities. J. Fluid Mech., 455:315–346,

2002.

[88] C. W. Rowley, I. Mezic, S. Bagheri, P. Schlatter, and D. S. Henningson. Spectral

analysis of nonlinear flows. J. Fluid Mech., 641:115–127, 2009.

[89] R. L. Sani and P. M. Gresho. Resume and remarks on the open boundary con-

dition minisymposium. International Journal for Numerical Methods in Fluids,

18(10):983–1008, 1994. ISSN 1097-0363. doi: 10.1002/fld.1650181006. URL

http://dx.doi.org/10.1002/fld.1650181006.

[90] V. Sarohia. Experimental and analytical investigation of oscillations in flows

over cavities. PhD thesis, California Institute of Technology, 1975. URL

http://resolver.caltech.edu/CaltechETD:etd-05032007-131245.

[91] P. Schmid and D. S. Henningson. Stability and Transition in Shear Flows. Springer,

New York, 2001.

[92] P. J. Schmid. Dynamic mode decomposition of numerical and experimental data.

Journal of Fluid Mechanics, 2010.

[93] P. J. Schmid. Dynamic Mode Decomposition. LS 2014-01. von Karman Institute

Lecture Series on ”Advanced Post-Processing of Experimental and Numerical Data”,

November 2013.

[94] P. N. Shankar and M. D. Deshpande. Fluid mechanics in the driven cavity. Annu.

Rev. Fluid Mech., 32:93–136, 2000.

[95] S. J. Sherwin and H. M. Blackburn. Three-dimensional instabilities of steady and

pulsatile axisymmetric stenotic flows. J. Fluid Mech., 533(297-327), 2005.

[96] D. Sipp. Open-loop control of cavity oscillations with harmonic forcings. Journal of

Fluid Mechanics, 708:439–468, 10 2012. ISSN 1469-7645.

BIBLIOGRAPHY 121

[97] D. Sipp and A. Lebedev. Global stability of base and mean flows: a general approach

and its applications to cylinder and open cavity flows. J. Fluid Mech., 593:333–358,

2007.

[98] V. Tatsumi and T. Yoshimura. Stability of the laminar flow in a rectangular duct.

J. Fluid Mech., 212:437–449, 1990.

[99] V. Theofilis. On steady-state flow solutions and their nonparallel global linear insta-

bility. In Dopazo, C, editor, Advances In Turbulence VIII, pages 35–38, Barcelona,

Spain, 2000. ISBN 84-89925-65-8.

[100] V. Theofilis. Globally-unstable flows in open cavities. 6th AIAA/CEAS Aeroacous-

tics Conference and Exhibit, Maui, HI, June 12-14 2000. AIAA Paper.

[101] V. Theofilis. Advances in global linear instability analysis of nonparallel and three-

dimensional flows. Prog. Aerosp. Sci., 39 (4)(4):249–315, 2003.

[102] V. Theofilis. Global linear instability. Annu. Rev. Fluid Mech., 43:319–352, 2011.

[103] V. Theofilis and T. Colonius. An algorithm for the recovery of 2- and 3-D BiGlobal

instabilities of compressible flow over 2-d open cavities. In AIAA Paper 2003–4143,

33rd Fluid Dynamics Conference and Exhibit, June 23 – 26, 2003, Orlando, FL, 2003.

[104] V. Theofilis and T. Colonius. Three-dimensional instablities of compressible flow

over open cavities: direct solution of the biglobal eigenvalue problem. 34th Fluid

Dynamics Conference and Exhibit, Portland, Oregon, June 28 – July 1, 2004 2004.

AIAA Paper 2004-2544.

[105] V. Theofilis, T. Colonius, and A. Seifert. Procedings afors/eoard/ercoftac sig-

33: Global flow instability control symposium i. Fundacion General Universidad

Politecnica, 2001.

[106] V. Theofilis, D. Barkley, and S. J. Sherwin. Spectral/hp element technology for flow

instability and control. Aero. J., 106:619–625, 2002.

[107] E. F. Toro. Riemann solvers and numerical methods for fluid dynamics : a practical

introduction. Springer, Berlin, New York, 1997.

[108] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll. Hydrodynamic

stability without eigenvalues. Science, 261(5121):578–584, 1993.

122 BIBLIOGRAPHY

[109] L. S. Tuckerman and D. Barkley. Bifurcation analysis for timesteppers. In E. Doedel

and L.S. Tuckerman, editors, Numerical Methods for Bifurcation Problems and

Large-Scale Dynamical Systems, volume 119, pages 543–466. Springer, New York,

2000.

[110] N. Vinha, F. Meseguer-Garrido, J. de Vicente, and E. V. A dynamic mode decom-

position of the saturation process in the open cavity flow. Aerospace Science and

Technology, 2014. submitted to Special Issue on ”Massively Separated Flows”.

[111] T. Wintergerste and L. Kleiser. Secondary stability analysis of nonlinear crossflow

vortices. In H. Fasel and W. Saric, editors, Proc. of the IUTAM Laminar-Turbulent

Symposium V, pages 583 – 586, Sedona, AZ, USA, 2000.

[112] S. Yamouni, D. Sipp, and L. Jacquin. Interaction between feedback aeroacoustic and

acoustic resonance mechanisms in a cavity flow: a global stability analysis. Journal

of Fluid Mechanics, 717:134–165, 2013. ISSN 1469-7645. doi: 10.1017/jfm.2012.563.

URL http://dx.doi.org/10.1017/jfm.2012.563.

[113] A. Zebib. Stability of viscous flow past a circular cylinder. J. Eng. Math., 21:

155–165, 1987.

instability analysis of incompressibleopencavity flows · instability analysis of...

Documents