reynolds number effects on disturbances in a laminar

REYNOLDS NUMBER EFFECTS ON DISTURBANCES IN A LAMINAR FREE JET

Bachelorarbeit von Pérez Roca, Sergio

durchgeführt am Institut für Aerodynamik und Gasdynamik Universität Stuttgart. Betreuer: apl. Prof. Ulrich Rist und Dr.-Ing. Oliver Schmidt

Stuttgart im April 2014

Pfaffenwaldring 21 · 70550 Stuttgart

Apl. Prof. Dr.-Ing. Ulrich Rist

Bachelorarbeit für Herrn Sergio Pérez Roca

„Reynolds number effects on disturbances in a laminar free jet“ dt.: „Reynoldszahl-Effekte auf Störungen in einem laminaren Freistrahl“

Figure 1: The two-dimensional laminar free jet after Schlichting (from [1]). The Reynolds number dependence of the analytical, planar, laminar jet solution after Schlichting (see Figure 1) is to be investigated by means of direct numerical simulation (DNS). First, the analytical solution is to be implemented as a base-state upon an appropriate grid for the use of the in-house DNS code NS3D [1]. Second, simulations at different Reynolds numbers are to be carried out and analyzed with respect to amplification of small perturbations. Initial perturbations are expected to occur naturally as the laminar base-state is no exact solution to the full compressible Navier-Stokes equations implemented in NS3D. Alternatively, random perturbations are to be artificially superimposed upon the laminar base-state in a more controlled scenario. The response of the flow is to be interpreted in terms of linear-stability characteristics, i.e. absolute or convective instability.

Literatur [1] F. M. White: Viscous fluid flow. McGraw-Hill, International Edition, S. 254, 2006. [2] A. Babucke: Direct Numerical Simulation of Noise-Generation Mechanisms in the Mixing Layer of a Jet. Dr. Hut, 2009.

Betreuer: Oliver T. Schmidt

Stuttgart, den 01. November 2013

Ausgabedatum: 01.11.2013 Abgabedatum: 30.04.2014

apl. Prof. Dr.-Ing. Ulrich Rist

I hereby declare that I have created this work completely on my own and with the help of my coordinator and used no other sources or tools than the ones listed, and that I have marked any citations accordingly. Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie Zitate kenntlich gemacht habe.

Stuttgart, April 2014

Sergio Pérez Roca

Acknowledgements Ich bedanke mich bei meinem Betreuer, Oliver, für die groÿe Hilfe, die er mir zu allen Fragen und Zweifeln geleistet hat, für seine Geduld mit kommunikativen Missverständnissen und für seinen groÿen Anspruch. Die Ergebnisse konnten immer besser sein. Auf dieser Weise haben wir gute Resultate erarbeitet. Ich bedanke mich auch bei meinem Mitbetreuer Björn Selent, der mir viel mit auf den Computer bezogenen Problemen geholfen hat. Vielen Dank auch an Professor Dr. Rist, der mir mit Bürokratie assistiert hat und der mir dazu verholfen hat, die Zulassung zum Master in Aerospace Mechanics and Avionics von ISAE in Toulouse zu bekommen. Meine Kommilitonen und Freunde aus dem Rechenraum, Mayrén Prieto, Martin Hartmann und Andreas Rumpf, sind auch sehr wichtig für mich gewesen, da sie immer bereit waren, mit technischen Zweifeln zu helfen. Danke für diese gemütliche Arbeitsatmosphäre. Letzlich ein groÿes Dankeschön an Melanie, die mir mit physikalischen Themen und deutschem schriftlichem Ausdruck beigestanden hat.

Übersicht Es sind mehrere Direct Numerical Simulations (DNS) von zweidimensionalen freien achen Luftstrahlen bei verschiedenen Reynolds-Zahlen

R MATLAB

Reb

ausgeführt worden, um ihren Einuÿ zu analysieren. Die mittels

berechnete Grundströmung, die den achen Strahl repräsentiert, wird durch ein selbstähn-

liches Modell von Schlichting (1933) charakterisiert, dessen Gültigkeit veriziert worden ist. Unterschiede in der Reynoldszahl ergeben sich aus der Änderung der charakteristischen Länge

b.

Dieses Parameter von

den Gleichungen von Schlichting repräsentiert die Breite des Strahls, in dem die axiale Geschwindigkeit zu 1% seines maximalen Wertes verfällt. maximale Geschwindigkeit

umax

Modizierungen dieses Parameters wirken sich nicht auf die

aus. Dadurch bleibt die Machzahl konstant, um dasselbe Kompressibil-

itätsverhalten zu behalten. Sieben DNS Berechnungen sind anhand des Eigencodes

NS3D

des Instituts

der Aerodynamik und Gasdynamik (IAG) an der Universität Stuttgart, Deutschland, durchgeführt wor-

Reb = 300 bis Reb = 2500. Ergebnisse für die axiale Geschwindigkeit und deren Fluktuation sind R und EAS3 ausgewertet worden. Das Resultat ist, dass sich stets zunächst Hilfe von Tecplot

den, von mit der

eine Instabilität in Form von mehreren Wellenpaketen ausbildet, und dann ein anschliessender zweidimensionaler Übergang zu Nichtlinearität stattndet. Je grosser

Reb ,

desto früher und intensiver treten

die beobachteten Eekte auf.

Abstract Several Direct Numerical Simulations (DNS) of two-dimensional free plane air jets at dierent Reynolds numbers

Reb

have been carried out, in order to analyse its inuence. The baseow employed to represent

the plane jet, calculated by means of

R , is characterised by following Schlichting's self-similar MATLAB

model (1933), whose validity has been veried. Dierences in Reynolds number among all the cases hinge upon a variant characteristic length

b.

This parameter in Schlichting's equations represents the width of

the jet in which the axial velocity decays to a 1% of its maximum value. Modications in this parameter have allowed to keep the maximum velocity

umax

constant at every case, and therefore the Mach number,

so as to maintain the same compressible behaviour. Seven DNS calculations have been performed by dint of the in-house code

NS3D of the Institute of Aerodynamics and Gas Dynamics (IAG) at the University of

Reb = 300 to Reb = 2500. Results in terms of axial velocity and its uctuation R and EAS3, with the outcome that instability generation the help of Tecplot

Stuttgart, Germany; from have been assessed with

in form of several wave packets and subsequent two-dimensional transition to non-linearity occur earlier and more intensely the greater

Sergio Pérez Roca

Reb .

I

Bachelor's Thesis IAG Universität Stuttgart

CONTENTS

CONTENTS

Contents Abstract

I

Table of Contents

II

Nomenclature

IV

List of Figures

VII

List of Tables

X

1 Introduction

1

1.1

The free plane jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Characteristics of a plane jet

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

1

1.3

Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

Jet Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.5

Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.6

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

6

2 DNS Code. Physical Model and Numerical Method 2.1

Direct Numerical Simulations

2.2

NS3D

2.3

Physical Model

2.4

Background

7

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

7

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

8

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

8

2.3.1

Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3.2

Properties of the Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Method

11

2.4.1

Spatial Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.4.2

Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.4.3

Parallelisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3 Computational Setup 3.1

15

Baseow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.1.1

Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.1.2

Reynolds number denition

16

3.1.3

Domain dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.1.4

Mesh selection

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

20

3.1.5

Fields denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.1.6 3.2

10

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

GUI

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

(Graphical User Interface) for the creation of baseow les with

R MATLAB

27

Calculation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.2.1

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.2.2

Damping zone or

3.2.3

NS3D's ns3d.i

Sergio Pérez Roca

sponge

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

31

le parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

II


CONTENTS

CONTENTS

3.2.4

NS3D's setup

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

41

3.2.5

Three-dimensional test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.2.6

Computational setup summary

43

le parameters

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

4 Postprocessing of results 4.1 4.2

45

Axial velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.1.1

45

Axial velocity elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Axial velocity perturbation

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

4.2.1

Axial velocity perturbation elds . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.2

Maximum perturbation amplitude at each

x-coordinate

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

4.2.3

Contours of maximum amplitudes at each time step with respect to

4.2.4

Maximum amplitude at each

4.2.5

Field maximum amplitudes at each time step

y -coordinate

for several

x-locations

x-coordinate

.

62 62 74 84

. . . . . . . . . .

95

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

102

5 Conclusions

104

Appendix 1: Simulation les links

110

References

113

Sergio Pérez Roca

III


CONTENTS

CONTENTS

Nomenclature ∗ ∈ 1, 2

derivation order

+, −

forward and backward biased FD, respectively

u¯0

root mean square (RMS) of the speed

∆ξ, ∆η

stepsizes in equidistant computational space

∆t

time step

η

self-similarity variable

ηK

Kolmogorov length scale

κ

heat capacity ratio

O

order of approximation

µ

dynamic viscosity

ν

kinematic viscosity

ω0

fundamental frequency of the simulation

Φ

any ow quantity

ψ

stream function of the jet

ρ

density

τ

turbulence time scale

τxx , τyy , τzz

normal stresses

τxy , τxz , τyz

shear stresses

F, G, H

ux vectors in

x- y -

and

T˜s

Sutherland temperature

x ˜0

dimensional start

˜

x

z -direction,

respectively

coordinate of the jet domain

dimensional quantity

ε

rate of kinetic energy dissipation

ϑ

thermal conductivity

ξ−η

computational space in

Sergio Pérez Roca

x−y

space

IV


CONTENTS

CONTENTS

∞

freestream value

lhs

left-hand side coecients

a

jet constant parameter

a−g

coecients of nite dierences

b

width of the jet for which

c

characteristic convective velocity of the problem

cp

heat capacity at constant pressure

cv

heat capacity at constant volume

D

nozzle or aperture diameter

E

total energy per volume

f

auxiliary function for the self-similar solution

h

mesh increments

J

momentum ux of the jet

j

index of gridpoint

Ku

velocity decay coecient

L

integral length scale of turbulence

l

equidistant time level, iteration number

Ma

Mach number

N

number of nodes of a mesh

Nx

total number of nodes in the

p

pressure

Pr

Prandtl number

Q

solution vector with conservative variables

qx , qy , qz

heat uxes in

x- y -

and

u(b/2) = 0.01umax

x

and characteristic length

axis

z -direction,

respectively

R

specic gas constant

Re

turbulent Reynolds number

Reb

Reynolds number of the jet ow based on the jet width

Sergio Pérez Roca

V

b


CONTENTS

CONTENTS

ReD

Reynolds number of the jet based on the nozzle diameter

T

temperature

t

time

u

streamwise velocity component

u0

perturbation of the axial velocity

Uc

local centreline mean velocity

Uo

bulk mean velocity

umax

reference axial velocity, maximum jet centreline velocity

v

normal velocity component

w

spanwise velocity component

x

streamwise coordinate

x∗

non-dimensional range of the transition zone in the damping zone tool

xv0

virtual origin of the jet

y

normal coordinate

z

spanwise coordinate

ZS _ende

last time step

ZS _output

number of output time steps

ZS _period

period of time steps

ZS _start

starting time step

CFD

Computational Fluid Dynamics

CFL

Courant-Friedrichs-Lewy or security factor number

DNS

Direct Numerical Simulation

EAS3 Ein-Ausgabe-System, IAG's programme FD

Finite Dierence

LES

Large-Eddy Simulation

MPI

Message Passing Interface

RANS Reynolds-Averaged Navier-Stokes

Sergio Pérez Roca

VI


LIST OF FIGURES

LIST OF FIGURES

List of Figures 1.1

Flow structure of a free jet [Yue Z., 1999]

1.2

Shear layer instabilities in a jet, [24]

3.1

Results for the

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

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

2 4

3.4

10bx10b domain . . . . . . Results for the 17.5bx10b domain . . . . . Results for the 85bx60b domain . . . . . . Mesh for the 10x10 domain at Reb = 1200

3.5

Axial velocity contours with 150x225 gridpoints per subdomain

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

3.6


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

22

3.7

Contours for the 200x275 gridpoint conguration

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

22

3.8


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

23

3.9

Meshing strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 3.3

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

18

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

19

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

19

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

20

3.10 Contour plots of axial velocity for the 24 subdomains at 3.11 Velocity eld for

Reb = 600

Reb = 1800

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

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

3.12 Axial (dimensional and non-dimensional) and transversal velocity contours for

Reb = 600

3.13 Mesh for

3.14 Sample of the

Reb = 600

.

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

R GUI MATLAB

Reb = 600

for

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

Reb = 1000 Outlow boundary condition detail with axial velocity contours and streamlines for Reb = 800 Characteristic free stream boundary condition for Reb = 600 . . . . . . . . . . . . . . . . . Free stream boundary condition for Reb = 600 . . . . . . . . . . . . . . . . . . . . . . . . .

3.15 Inow boundary condition detail with axial velocity contours and streamlines for 3.16 3.17 3.18

sponge

3.19 Comparison between nolation at

Reb = 600

and

sponge

3.20 Contours of axial velocity perturbation with

sponge

Reb = 600

Reb = 600

25 26 26 28 29 29 30 30 32

of gain 0.5 at the nal 5% points of

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

3.21 Contours of axial velocity uctuation without random perturbation, with 0.5 by normal calculation at

23 24

approaches of gain 3 last 5% by normal calcu-

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

the domain by normal calculation at

21

sponge

33

of gain

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

33

3.22 Comparison between randomly perturbed and not perturbed approaches of gain 0.5 by disturbance calculation at 3.23 Comparison between noat

Reb = 600

Reb = 600

sponge

3.25

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

sponge

sponge

Reb = 600

and

sponge

34

approaches of gain 0.5 by normal calculation

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

3.24 Comparison between notion at

and

34

approaches of gain 0.5 by disturbance calcula-

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

35

. . . . . . . . . .

36

R at Reb = 2500 Mean axial velocity at centreline obtained with Tecplot

3.26 Comparison between normal and disturbance calculation for axial velocity uctuation after 0.05 jet periods at

Reb = 600

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

37


Reb = 600

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

38

3.28 Comparison between normal and disturbance calculation for axial velocity uctuation after 1 jet period at

Reb = 600

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

38


Sergio Pérez Roca

Reb = 600

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

VII

39


LIST OF FIGURES

LIST OF FIGURES


Reb = 600

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

40

3.31 Comparison between normal and disturbance calculation for axial velocity uctuation after 2.8 jet periods at 3.32 3.33

Reb = 600

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

ns3d.i edition sample at Reb = 2500 setup edition sample at Reb = 2500 .

40

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

41

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

42

3.34 Contour plots of axial velocity with streamlines and isosurfaces for the 3D approach at

Reb = 500

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

Reb Reb Reb Reb Reb Reb Reb

= 300 . = 600 . = 800 . = 1000 = 1200 = 1800 = 2500

4.1

Contour plots of axial velocity at dierent time steps, at

4.2


4.3


4.4


4.5


4.6


4.7


4.8

Contour plots of axial velocity at dierent time steps in exponential scale, at

4.9

Contour plots of axial velocity at dierent time steps in exponential scale, at

43

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

46

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

47

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

48

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

49

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

50

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

51

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

52

Reb Reb Reb Reb Reb Reb Reb

= 300 = 600 = 800 = 1000 = 1200 = 1800 = 2500

.

55

.

56

.

57

.

58

.

59

.

60

.

61

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

65

4.10 Contour plots of axial velocity at dierent time steps in exponential scale, at 4.11 Contour plots of axial velocity at dierent time steps in exponential scale, at 4.12 Contour plots of axial velocity at dierent time steps in exponential scale, at 4.13 Contour plots of axial velocity at dierent time steps in exponential scale, at 4.14 Contour plots of axial velocity at dierent time steps in exponential scale, at

4.17 Contour plots of the absolute value of axial velocity uctuation at dierent time steps, at

Reb = 300


Reb = 600

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

67


Reb = 800

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

68


Reb = 1000

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

69


Reb = 1200

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

70


Reb = 1800

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

71


Reb = 2500

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

4.25 Maximum absolute values of axial velocity perturbation with respect to logarithmic scales at

Reb = 300 Reb = 600 Reb = 800

Sergio Pérez Roca

Reb = 1000

x

77

in natural and

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

VIII

76

in natural and

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


x

75

in natural and

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


x

72

in natural and

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


x

78


LIST OF FIGURES

LIST OF FIGURES


Reb = 1200 Reb = 1800 Reb = 2500

x x

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

4.33 Contours of maximum absolute values of axial velocity perturbation at each wiht respect to time at

Reb = 300 Reb = 600

wiht respect to time at

Reb = 800


Reb = 1000


Reb = 1200


Reb = 1800


Reb = 2500

90 91

x-coordinate

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

4.40 Group velocities with respect to the Reynolds number

89

x-coordinate

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

4.39 Contours of maximum absolute values of axial velocity perturbation at each

88

x-coordinate

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


87

x-coordinate

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


86

x-coordinate

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


83

x-coordinate

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


81

x-coordinate

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

4.34 Contours of maximum absolute values of axial velocity perturbation at each wiht respect to time at

80

in natural and

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

4.32 Non-dimensional jet period with respect to the Reynolds number

79

in natural and

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


in natural and

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


x

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

92 94

4.41 Logarithm of the maximum absolute values of axial velocity perturbation with respect to

y

at

Reb = 300

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

95


y

at

Reb = 600

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

96


y

at

Reb = 800

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

97


y

at

Reb = 1000

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

98


y

at

Reb = 1200

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

99


y

at

Reb = 1800

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

100


y

at

Reb = 2500

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

101

4.48 Global maximum absolute values of velocity perturbation amplitudes with respect to jet periods in logarithmic scale for the whole Reynolds number span

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IX

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

102


LIST OF TABLES

LIST OF TABLES

List of Tables 3.1

Summary of the computational setup of the common frame

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

44

4.1

Transition in jet periods at each Reynolds number observed in velocity contours . . . . . .

53

4.2

Transition in jet periods at each Reynolds number observed in maximum amplitude versus

x

plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.3

Non-dimensional jet period at each Reynolds number . . . . . . . . . . . . . . . . . . . . .

83

4.4

Wavelenghts, periods, phase velocities, wavenumbers and frequencies at each Reynolds number

4.5

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

84

Group velocities at each Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

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1

INTRODUCTION

1 Introduction 1.1 The free plane jet The free plane jet represents one of the two basic types of free symmetrical jet ows. The other is the round or axisymmetric one.

Free jets have become a standard for research into the physics of turbu-

lent uid ow, since they are commonly used for the evaluation of physical models. At suciently high Reynolds numbers the jet will be turbulent. Moreover they stand for a valuable resource for many engineering applications (jet propulsion, combustion chambers, etc.). In spite of the perhaps more limited direct applications of the plane type in comparison with the round, there are plenty of applications of planar ows in general. Regarding the aerospace sector, some propulsion units are based on these ows, such as ramjets or scramjets. Another application more common in daily life is air curtain devices. These technologies are usually integrated in the popular devices employed for energy saving in public buildings, refrigeration cycles or for air quality control in many kinds of industries, such as food or electronics. With regards to their biological applicability, these devices are used in the reduction or deletion of chemical compounds, scents, microorganisms, insects, dust, humidity and radioactive particle transport. They are also employed with the aim of scattering hazardous smoke in re safety systems for underground tunnels. Furthermore, practical applications in the elds of heating, ventilation and air conditioning are being currently developed. There is great interest in the transport and mixing processes of scalars in turbulent shear ows due to their importance in the propagation of contaminants in environmental ows, as well as the wide range of applications involving turbulent combustion. Therefore, research of plane jets brings about benecial application developments.

1.2 Characteristics of a plane jet Free jets can be dened as a pressure driven unrestrained ow of a uid into a static environment. Seeing that a uid boundary cannot maintain a pressure gradient across it, the subsonic jet boundary is a free shear layer in which the static pressure keeps constant throughout. The boundary layer at the exit of the nozzle evolves as a free shear layer, merging with the ambient uid thereby entraining the ambient uid in the jet stream. Thus, the mass ow at any cross section of the jet progressively increases. So as to conserve momentum, the jet centreline velocity shrinks with downstream development. Concretely, a plane jet is by denition a two-dimensional ow with a prevailing mean ux in the streamwise

x-direction, spread due to entrainment in the transversal y -direction and periodic distribution in z -direction if we consider the third dimension. Therefore, ow parameters in this spanwise direction are a periodic function of the z -coordinate. Practicable plane jets emerge from rectangular aperor axial

the spanwise

tures or nozzles of high aspect ratios. In order to consider such an actual jet as a plane one, it must be restrained inside two parallel walls appended to the short sides of the nozzle and situated in the x-y plane.

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1

INTRODUCTION

1.2

This kind of conguration permits the jet to disperse in the crosswise

Characteristics of a plane jet

y -direction

only, in practical terms.

In this thesis we will not take account of three-dimensional eects, since the periodic boundary conditions maintain the ow practically two-dimensional, allowing little entrainment in the spanwise direction. It is certain so, that even three-dimensional plane jets present a time-averaged mean velocity eld that remains two-dimensional, matching its denition of two-dimensional ow. The key parameters of a plane jet are the following. Its characteristic velocity is frequently dened in the literature as the area-averaged exit velocity at the nozzle, also called the bulk mean velocity,

Uo .

Nevertheless, in this study we will dene this characteristic speed as the centreline speed at the nozzle exit. Owing to the aforesaid entrainment, there is a far-eld progressive reduction in the centreline velocity, found to agree with:

where

Uc

Uo Uc

2

= Ku ∗

is the local centreline mean velocity,

the nozzle, at some distance before the exit;

D

xv0

x − xv0 , D

(1.1)

is the virtual origin, which is considered to be inside

is the nozzle width and

Ku

is the coecient that models

the velocity decay, as Ravinesh C. Deo states, [1].

Figure 1.1:

Flow structure of a free jet [Yue Z., 1999]

With respect to the jet spatial evolution, it is relevant to point out the main regions of which it consists, plotted in Figure 1.2. The initial region occurs between the jet origin and around four to six aperture diameters downstream. This is designated as the potential core, where the average velocity is virtually uniform. All over this conical potential core, the mean centerline velocity

Uc

is close upon

Uo .

After this

region there is the interaction or transition region, regularly enclosed between six and twenty diameters downstream.

In this stage, momentum transport is facilitated through large-scale vortices interaction,

since shear layers from both sides merge. Here, the velocity decay can be approximated as proportional

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1

to

INTRODUCTION

x−0.5 .

1.3

Self-similarity

At a greater distance downstream, the ux is mathematically expected to become self-similar

at dierent values of x and the main parameters such as decay, spreading rates, uctuation intensity, etc. asymptote to converged values, the velocity decay being proportional to

x−1 .

After the self-similar

region some authors consider the termination region, in which velocity decays quickly. The mechanisms governing this stage are currently not understood properly.

1.3 Self-similarity It is worth it to make clear the concept of self-similarity, which will be present throughout this thesis, being an important characteristic of free jets. Self preservation or self-similarity is dened as the ow state given when the proles of velocity (or any other eld) can be assumed congruent by plain scale coecients depending exclusively upon one transformed coordinate. As a consequence, the governing equations of jet ow can be simplied to ordinary dierential equation conguration. Besides, a ow is also considered to be self-similar if there exist solutions to its dynamical equations and boundary conditions for which, all over its evolution, every term has the identical relative value at the corresponding relative location. Hence, self preservation indicates that the ow has acquired a sort of equilibrium in which all of its dynamical inuences develop in unison, and relative dynamical readjustment is no longer required. Self-similarity is for this reason an asymptotic state a ow reaches after its internal adaptation is concluded.

There

exist dierent types of self-similarity states, as Premnath R. asserts, [24]. On the one hand, ows can be entirely self preserving at all orders of the turbulence moments and at all scales of motion. On the other hand, ows can be partially self preserving, in the sense that they are so only at the level of the mean momentum equations, or until only certain orders of the turbulence moments or at certain scales.

1.4 Jet Instabilities It is also interesting to explain the main jet instabilities, that have been proved to occur in several studies mentioned in section 1.5, when increasing the Reynolds number. The origin and development of these instabilities will be a major research point all over this thesis. After some axial development, and with dependence on the initial velocity prole, the jet uid emanating from the opening attains ow perturbations in the shear layer, [24]. In three dimensions, these perturbations will roll up to create vortices which augment in size and intensity with the longitudinal remoteness. The vortices will aect the entrainment of the uid medium and the merging of the ambient uid and the jet uid. These vortex interactions will culminate in the transition of the ow to turbulent regime. Nevertheless, in two dimensions, transition to turbulence does not really occur, since the third component is essential in the turbulence denition. Concerning this phenomenon in 2D, we will simply treat transition to non-linearity.

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1

INTRODUCTION

1.5

Figure 1.2:

Literature review

Shear layer instabilities in a jet, [24]

1.5 Literature review Due to the important aforementioned applications, plane jets have received signicant research attention both in experimental and numerical investigations.

Experimental data necessary to analyse the main

parameters of the plane jet as well as to procure verication in the elaboration of turbulence models. Numerical investigations generally consist of DNS and LES simulations. If we move back to the mathematical conception of free jets, Schlichting (1933), [2], whose self-similar solution is the initial ow condition used in this thesis, was one of the rst researches who analised free jets in depth, developing mathematical models for planar jets.

In his most characteristic and reputed

paper, he studied the spread of a jet emanating from an aperture into a stationary uid environment for the two types of jets mentioned. The closed-formed integration of the governing equations was made possible by his models, by means of reducing it to an axial-symmetrical problem. He conceived of the plane jet as an emerging ux from a long narrow opening, leading to a two-dimensional ow eld. Therefore, Schlichting devised a numerical method with which to integrate the leading equations, obtaining a interestingly simple solution for these types of ows. Subsequently, Bickley (1937), [3], broadened the work of Schlichting (1933).

By performing numerical

calculations of the mass ow rates of a plane jet with the continuity equation, he discovered the presence of entrainment of the surrounding uid as the jet propagates axially. This contributes to a reduction in the longitudinal centreline speed. The main consequence of this phenomenon is the invariability of the momentum. Probably the rst experimental study of a plane jet, driven by Forthman (1934), [4], consisted in measuring the average velocity of the ow along a span of 25 aperture diameters downstream by means of total head pressure tubes. Nevertheless this range was considered to be scarce and not sucient to consider total development of the jet. Further and more complete investigations, carried out by Miller and Comings (1957), [5], employed hot wire anemometry over 40 diameters so as to obtain the mean, root-mean-square (rms) and static pressure distributions. Furthermore, Van Der Hegge Zijnen (1958), [6], performed an experimental study of transversal average velocity and Reynolds stresses utilizing not only hot wire anemometry but also total

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1

INTRODUCTION

1.5

Literature review

head tubes along the distance Forthman (1934) considered (25 diameters). He developed an experimental setup with two aperture aspect ratios and discovered variations in the progagation and weakening rates of the centreline mean velocity. It could also be drawn from his analysis that the jet could be well modeled with a plane wake and acquired self-similarity at approximately 30 nozzle diameters downstream. Also using hot wire anemometry, which is by far the most used method in jets experiments, Heskestad (1965), [7], studied sharp-edged orice nozzles. He used a longitudinal span of 160 nozzle diameters and came out with several longitudinal and transversal properties: energy budget, atness factors and intermittency. The inuence of Reynolds number on axial turbulence intensity, one of the main objectives of this thesis, was barely evaluated. Even more investigations followed this one, conducted by Jenkins and Goldschmidt (1973), [8], Gutmark and Wygnanski (1976), [9], and Thomas and Goldschmidt (1986), [10]. In general, these experimental setups measured a plane jet at various baseow distributions, helping to understand deeplier the properties of these ows. Each of these studies obtained a dierent start of the selfsimilar solution, also named asymptotic behaviour. These variations coincide with the results of George (1989), [11], in other words, initial conditions are determinant in the evolution of the ow eld of a free jet. More recently, with the improvement of computing power, some numerical studies on free jets have been done, especially on round jets but also on plane ones. ited.

Comte

et al.

Nevertheless, the amount of them is quiet lim-

(1989), [12], carried out temporal simulations of the totally developed region of a

two-dimensional jet to study the eects of dierent initial conditions. Stanley & Sarkar (1997), [13], contrasted two-dimensional strong and weak jets with three-dimensional jets and came upon that inaccurate mean velocity proles can be obtained when simulating two-dimensional strong planar jets, therefore they are usually avoided in real research. Dai, Kobayashi & Taniguchi (1994), [14], carried through the rst simulations of a subsonic threedimensional spatially evolving, planar jet employing large-eddy simulation (LES). Their results matched convincingly the experimental data in their average proles. Nonetheless their resulting self-similar turbulence intensities exceded those data in a 40%. The direct numerical simulations (DNS) by Boersma

et al.

(1998), [15], which were carried out for round

jets in this case, agreed with the results of George(1989), in that the inuence of the initial conditions is vital when trying to obtain self-similar solutions. They demonstrated that with a correct scaling of the problem, in which dependence on the initial conditions is included, brought about agreement with the self-similarity hypothesis. However, neither of these investigations took into account the eects of the chosen LES model. Le Ribault, Sarkar & Stanley (1999), [16], contrasted in depth the Smagorinsky, dynamic Smagorinsky and dynamic mixed LES models in simulations of the near-eld region of planar turbulent jets and stumbled on a valid agreement for the latter two models, not only with direct numerical simulation (DNS) but also with higher Reynolds number laboratory experiments. Stanley

et al.

(2001), [17], calculated one of the rst direct numerical simulations (DNS) of a plane jet.

They demonstrated that DNS, considering a high-order space and time accuracy and pertinent boundary conditions, represents a precise tool to analyse the spatial evolution of a plane jet, since their results tted

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1

INTRODUCTION

1.6

Objectives

satisfactorily with experimental data. M. Klein

et al.

(2003), [18], developed a direct numerical simulation of plane turbulent jets at moderate

Reynolds numbers. They attended the inuences of the Reynolds number and the inow conditions on the evolution of the jet. It is important to point out that this is the most similar paper to this thesis found by the author, in terms of objectives and instruments employed. Regarding the rst parameter, they observed that the ow is not independent of the Reynolds number but almost reaches a converged state at Re=6000, based on the initial mean velocity and the nozzle diameter. In this thesis we will dene a slightly dierent Reynolds number, but similar in concept. Moreover, they conrmed that the inuence of the inow conditions on the jet characteristics is really strong and long-living. In summary, a conclusion which has been drawn in various studies is the fact that the initial ow and boundary conditions play a determinant role in the spatial evolution of free jets. Therefore, as interesting and intriguing as this observation seems, this will be one of the key points all over this thesis.

1.6 Objectives After introducing the main topic of this work, it is relevant to state the principal objectives of this Bachelor's Thesis:

•

Study the validity of Schlichting's plane jet self-similar solution (1933) as a baseow for a DNS calculation at low Reynolds numbers.

•

Obtain an optimum 2D simulation and visualisation of the free plane air jet via

NS3D

DNS code and

EAS3

toolkit, and

R, Tecplot

R , IAG's MATLAB

in terms of observability of all development

phases, accuracy of simulation, and similarity to reality.

•

Analyse the eects of Reynolds number on the jet development, maintaining a constant Mach number, including instability generation in laminar regime until transition to non-linearity.

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2

DNS CODE. PHYSICAL MODEL AND NUMERICAL METHOD

2 DNS Code. Physical Model and Numerical Method 2.1 Direct Numerical Simulations In order to perform a CFD (Computational Fluid Dynamics) simulation of a free plane jet, the type of calculation selected has been the Direct Numerical Simulation (DNS). These sort of codes are intended to represent exactly the reality of the uid mechanics, containing no modelling. The whole Navier-Stokes equations are simulated numerically, resolving the entire spectrum of turbulent eddies. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov scales), up to the integral scale most of the kinetic energy, as told in

CFD Online, [19].

L,

associated with the motions containing

The Kolmogorov scale,

ηK ,

is expressed as:

ηK = (ν 3 /ε)1/4

(2.1)

ν being the kinematic viscosity and ε the rate of kinetic energy dissipation. Regarding the integral scale L, it is relevant to highlight its usual dependence on the spatial scale of the boundary conditions. So as to full these resolution necessities, the amount N of points on any of the mesh axes with increments h, has to be:

N h > L,

(2.2)

with the eect that the integral scale is included in the computational domain, without forgetting:

h ≤ ηK ,

(2.3)

with the aim of resolving the Kolmogorov scale. Since

3 ε ≈ u¯0 /L,

u¯0

(2.4)

being the root mean square (RMS) of the speed. The implications of the preceding associations are

that a 3D DNS demands an amount of mesh nodes

N3

satisfying

N 3 ≥ Re9/4 where

Re

(2.5)

is the turbulent Reynolds number

Re =

u0 L . ν

(2.6)

Thus, the greater the Reynolds number the huger the memory storage required in a DNS. Furthermore, seeing the extensive memory enforced, the time integration of the solution is to be done by an explicit method. This implies that so as to be accurate, the CFL (Courant-Friedrichs-Lewy) or security factor

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2


2.2

number must be equal or less than the unit, as we will explain in section 3.2.3. simulated is as a rule proportional to the turbulence time scale

τ

NS3D

Background

The entire time step

expressed as:

L . u0

τ=

(2.7)

h must be of the order of ηK , the total sum of L/(CF LηK ). Separately, from the aforementioned

Coupling these expressions, and owing to the fact that time-integration intervals must be proportional to

Re, ηK

relations for

and

L,

it results:

L ∼ Re3/4 , ηK

(2.8)

with the consequence that the number of time intervals increases as a power function of the Reynolds number too. The amount of oating-point operations needed to perform the simulation is proportional to the number of mesh nodes and the number of time intervals, leading to a growth in the number of operations as

Re3 .

Consequently, the computational expense of DNS is very elevated, still at low Reynolds numbers. This is the main reason why the usage of this method has not spread over the industrial sector. Nonetheless, direct numerical simulation is a powerful instrument in basic research in turbulence. By means of DNS it is conceivable to carry out "numerical experiments", achieving a better comprehension of the nature of turbulence. Hence, turbulence models for practical applications can be deduced, for instance small scale models for Large Eddy Simulation (LES) and closure models for the RANS (Reynolds-Averaged NavierStokes equations). This processes are performed making use of "a priori" tests, taking the baseow data for the model from a DNS simulation, or by "a posteriori" tests, comparing output values produced by the model with those simulated by DNS.

2.2 NS3D

NS3D

Background

is a numerical method for direct numerical simulation (DNS) of the compressible Navier-Stokes

equations.

It was written by Andreas Babucke within his PhD Thesis on DNS of Noise-Generation

Mechanisms in the Mixing Layer of a Jet, [20], at the at

Universität Stuttgart.

Institute of Aerodynamics and Gas Dynamics (IAG )

It is still used at this Institut for DNS of subsonic, transonic and supersonic

ows and further developed. In this thesis we will take advantage of this powerful tool also within

IAG

facilities. Apart from the DNS Code core, it is accompanied by plenty of postprocessing tools based on the

EAS3

tool set, which represent a practical way of dealing with the generated data.

2.3 Physical Model Despite being a DNS, some modelling has to take place up to a certain extent, in terms of integration domain and partial dierential equations. This physical model is explained in the following paragraphs following the development of Babucke, [20].

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2.3

Physical Model

2.3.1 Governing Equations Although the conguration of the domain in this thesis will be two-dimensional, the unsteady compressible Navier-Stokes equations are formulated in the programme three-dimensionally and in a conservative way, attaining the solution vector with conservative variables

Q:

Q = (ρ, ρu, ρv, ρw, E)T including the density

ρ,

(2.9)

the three mass uxes related to the three velocity components

the total energy per volume

E,

u, v

and

w

and

commonly expressed as:

Z

ρ 2 u + v 2 + w2 . 2 cv being the non-dimensional heat capacity at constant volume and T the temperature. E=ρ

cv dT +

(2.10) The N-S equations

can be expressed in vector notation:

∂Q ∂F ∂G ∂H + + + =0 ∂t ∂x ∂y ∂z

(2.11)

F, G, H being the ux vectors and t the time:    F=      G=      H=   with

p

ρu 2 ρu + p − τxx ρuv − τxy ρuw − τxz u(E + p) + qx − uτxx − vτxy − wτxz



ρv ρuv − τxy ρv 2 + p − τyy ρvw − τyz v(E + p) + qy − uτxy − vτyy − wτyz



ρw ρuw − τxz ρvw − τyz ρw2 + p − τzz w(E + p) + qz − uτxz − vτyz − wτzz



as the pressure and the normal stresses

τxx

µ = Re

τxx , τyy

and

Sergio Pérez Roca

    

    

(2.12)

(2.13)

(2.14)

τzz :

4 ∂u 2 ∂v 2 ∂w − − 3 ∂x 3 ∂y 3 ∂z

4 ∂v 2 ∂u 2 ∂w τyy − − 3 ∂y 3 ∂x 3 ∂z µ 4 ∂w 2 ∂u 2 ∂v τzz = − − , Re 3 ∂z 3 ∂x 3 ∂y µ = Re

    

(2.15)

9

(2.16)

(2.17)


2


shear stresses

τxy , τxz

and

qx , qy

and

τxy

µ = Re

τxz

µ = Re µ Re

∂u ∂v + ∂y ∂x

∂u ∂w + ∂z ∂x

∂v ∂w + ∂z ∂y

(2.18)

(2.19)

(2.20)

qz : qx = −

ϑ ∂T · 2 (κ − 1)ReP rM a∞ ∂x

(2.21)

qy = −

ϑ ∂T · 2 (κ − 1)ReP rM a∞ ∂y

(2.22)

ϑ ∂T · , 2 (κ − 1)ReP rM a∞ ∂z

(2.23)

qz = − with

Physical Model

τyz :

τyz = and heat uxes

2.3

ϑ as a non-dimensional thermal conductivity, κ as the heat capacity ratio (κ = cp /cv = 1.4 for air), µ P r as the Prandtl number and M a as the Mach number. The rest

as a non-dimensional dynamic viscosity,

of variables, such as the velocity eld, will be dened throughout section 3, in the computational setup explanation. They are all required to be non-dimensional, since that is the mode in which computations are performed with

NS3D,

[20]. The specied denotation by the programme designers, also followed all

over this thesis, consists in dimensional values having a tilde ( ˜ ), non-dimensional being the plain letter, and the subscript ∞ as a freestream or reference initial value. Due to our two-dimensional approach, all the terms related to the

z

axis such as

w, τxz

or

∂/∂z

will be

ignored.

2.3.2 Properties of the Fluid The uid is presumed to be a non-reacting ideal gas following the equation of state being taken up later in section 3.1.5:

pnon−dim = p =

ρT , κM a2

(2.24)

in non-dimensional terms. The heat capacities at constant pressure (cp ) and volume are established as constant:

cp =

1 1 2 , cv = (κ − 1). M a∞ κ(κ − 1). M a∞ 2

(2.25)

The viscosity is dependent on temperature according to the Sutherland law, [21], for temperatures greater than the Sutherland temperature

Sergio Pérez Roca

T˜s = 110.4K : 10


2


µ ˜(T ) = µ ˜0 (T˜0 ) · T 3/2 · where

µ ˜0 (T˜0 = 280K) = 1.735 · 10−5 kg/(ms)

and

2.4

Numerical Method

1 + Ts , T + Ts

Ts = T˜s /T˜∞

(2.26)

is dened as the non-dimensional Suther-

land temperature. For lower temperatures the viscosity is considered to follow a linear dependence on temperature. The Prandtl number is kept to 0.71 since the uid in study is air. It is worth it to dene this non-dimensional property of the uid:

c˜p µ ˜∞ , ˜ ϑ∞

Pr = with

c˜p , µ ˜∞

and

ϑ˜∞

being the dimensional heat capacity at constant pressure, dynamic viscosity and

thermal conductivity respectively. The dimensional

c˜p

c˜p = cp where

umax

(2.27)

is equal to:

umax 2 , T˜∞

(2.28)

is our reference axial velocity, dened in section 3.1.1.

2.4 Numerical Method The numerical method employed is explained in the following paragraphs, thanks to Babucke, [20], and the useful

TRANSIWIKI

of

IAG, [22].

2.4.1 Spatial Discretisation The spatial discretisation in

x

and

y

axes is performed by Compact Finite Dierences (FD) of 6

Subdomain Compact Finite Dierences or Explicit Finite Dierences of

ξ−η

th 8

th

order,

order on the equidistant

computational grid. Dealiasing is done by alternating forward-backward weighted dierences for

convective terms.

nd

2

derivatives are directly calculated, not as usual, being related to twice the rst

derivative. To solve the tridiagonal systems of equations of compact dierences in several subdomains, a complex parallelisation scheme is observed. In this thesis, Compact FD, the simplest approach, are used, since the mesh is not very large, and calculations are performed within a reasonable time. It is true that Subdomain Compact FD would be a quicker option but they also demand more computational resources at a time, and for a 2D-simulation this is not especially required.

Compact Finite Dierences Finite dierence methods are frequently applied in solving dierential equations. A FD or Padé scheme is compact in the sense that the discretised formula involves at most nine point stencils comprising a node in the centre about which dierences are taken, as Lele contemplated in their work, [23]. Moreover, great order of accuracy can be achieved, taking into account the truncation error. In general, the method has proved robustness, eciency and accuracy for most computational uid dynamics (CFD) applications. Consequently they are also adequate for such an accuracy-demanding code as DNS.

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2.4

Numerical Method

The rst derivatives of the convective terms are carried out by means of alternating forward-backward weighted dierences. For the

21

dierence, dened in the programme and used throughout this thesis,

has the following expressions (for

1 ∂Φ 3 · + · 5 ∂ξ j−1,+ 5 3 1 ∂Φ + · · 5 ∂ξ j−1,− 5

∆ξ, ∆η

∂Φ 1 + · ∂ξ j,+ 5 1 ∂Φ + · ∂ξ j,− 5

being identic):

−1 · Φj−2 − 19 · Φj−1 + 11 · Φj + 9 · Φj+1 ∂Φ = ∂ξ j+1,+ 30 · ∆ξ −9 · Φj−1 − 11 · Φj + 19 · Φj+1 + 1 · Φj+2 ∂Φ = , ∂ξ j+1,− 30 · ∆ξ

the rst being suitable for forward dierences and the second for backward ones. tive variables of the problem,

j

the index of gridpoint and

+, −

Φ

(2.29)

(2.30)

is any of the convec-

the forward and backward biased FD,

respectively. The weighting changes between each Runge-Kutta sub-step, explained in the next paragraph. The dierence stencils are changed so that every combination of forward-backward weighting occurs in the

z -direction

and

and no preferred direction exists. In the input le

dierences for the spatial directions can be specied,

21

x-, y -,

ns3d.i, addressed in section 3.2.3, the

in this case. There are several options available

for these coecients, but this was considered as appropriate due to its simplicity. The rst derivatives of the viscous terms will be discretised in the following way, also in Compact FD of

th

6

order:

−1 · Φj−2 − 28 · Φj−1 + 28 · Φj+1 + 1 · Φj+2 ∂Φ ∂Φ ∂Φ +3· +1· = 1· ∂ξ j−1 ∂ξ j ∂ξ j+1 12 · ∆ξ The second derivatives will be directly calculated, as said above.

For 6

th

(2.31)

order the dierence stencil

corresponds to:

2·

3 · Φj−2 + 48 · Φj−1 − 102 · Φj + 48 · Φj+1 + 3 · Φj+2 ∂ 2 Φ ∂ 2 Φ ∂ 2 Φ + 11 · + 2 · = ∂ξ 2 j−1 ∂ξ 2 j ∂ξ 2 j+1 4 · ∆ξ 2

(2.32)

Dierentiation at the boundaries is carried out by a one-sided explicit stencil such as:

a · Φj=1 + b · Φj=2 + c · Φj=3 + d · Φj=4 + e · Φj=5 ∂ ∗ Φ = , ∗ ∂ξ j=1 ∆ξ ∗

(2.33)

and at the point next to the boundaries the following biased compact FD is used:

a · Φj=1 + b · Φj=2 + c · Φj=3 + d · Φj=4 + e · Φj=5 + f · Φj=6 + g · Φj=7 ∂ ∗ Φ ∂ ∗ Φ blhs · + clhs · = , ∂ξ ∗ j=2 ∂ξ ∗ j=3 ∆ξ ∗ (2.34) where the coecients

a−g

depend on the boundary condition chosen, explained in section 3.2.1;

∗ ∈ 1, 2

denotes the the derivation order, and the subscript lhs designates the left-hand side coecients. For the right-hand ones the direction of the stencil needs to be reversed.

Sergio Pérez Roca

12


2


2.4

Numerical Method

2.4.2 Time Integration 4th order Runge-Kutta Method The time integration is performed applying the classical four-step Runge-Kutta scheme. The main advantages of this method are its high accuracy (O

4)

and robustness facing oscillation and diusion problems.

The Runge-Kutta method structure comprises 4 steps: a predictor and a corrector step for the intermediate time step

t + ∆t/2

and for the new time step

l+ 1 Qk 2

l+ 1 Qk 2

=

∗ Qkl+1

Ql+1 k

=

Qlk

+ ∆t ·

1 ∂Q l + ∆t · 2 ∂t +/−

(2.35)

1∗ 1 ∂Q l+ 2 + ∆t · 2 ∂t −/+

(2.36)

1 ∂Q l+ 2 + ∆t · ∂t +/−

(2.37)

∗

=

Qlk

Qlk

=

t + ∆t:

Qlk

1∗ 1 ∗! 1 ∂Q l 1 ∂Q l+ 2 1 ∂Q l+ 2 1 ∂Q l+1 · + · + · + · , 6 ∂t +/− 3 ∂t −/+ 3 ∂t +/− 6 ∂t −/+

(2.38)

Q denoting the vector of the conservative variables (ρ, ρu, ρv, ρW, E ) and ∆t the increment in time. time level is specied by the time step from level

l and the intermediate values of l to l + 1. At each intermediate

The

∆t corresponding to derivatives ∂ Q/∂t and so the

the predictor step by *, level, the time

spatial derivatives are recalculated. The direction of forward-backward alternating dierences is indicated by +/− . From time step to time step, the alternation is changed, so that all combinations appear in the

x-, y -

and

z -direction

and a preferential direction is avoided.

Disturbance Computation Until now we have described the normal or regular calculation.

The code also includes the option of

a disturbance or perturbation calculation, which assumes the initial conditions ow or baseow to be a steady-state solution of the Navier-Stokes equations. Since a disturbance formulation of the Navier-Stokes equations is not achievable due to their non-linearity, a new approach is implemented, [20]. It consists in the computation for the baseow of time derivatives of the conservative variables for each one of the combinations of the forward-backward biased discretisation. values

∂Q0 /∂t

Throughout these simulations the stored

are substracted from the calculated time derivatives. Hence, the time derivatives of the

baseow represent the source term, constraining the determined baseow to be a steady solution.

By

means of the same discretisation for the source term and the calculation, the procedure is analogous to a genuine disturbance formulation. With regards to boundary conditions prescribing time derivatives, this method is carried out likewise to the respective boundary.

If ow elds are determined, the divergence of the baseow is saved and

afterwards substracted from the recently calculated values. The activation of this type of calculation is performed in the

Sergio Pérez Roca

13

ns3d.i

le explained in section 3.2.3


2


2.4

Numerical Method

2.4.3 Parallelisation As explained later in section 3.1.4, a descomposition of the domain in 24 subdomains is implemented in order to enhance the rapidity of calculations. Hence, we make use of the parallelisation algorithm of

NS3D,

[20]. In this code, a hybrid parallelisation of both MPI (Message Passing Interface) and shared

memory parallelisation is conceived. The key aspect of this concept is the domain descomposition in the

x-y

plane with data exchange using MPI. It allows to run a simulation on several nodes. This type of

descompositions facilitates the use of more complex geometries. Message passing between neighbouring subdomains is only needed for calculating spatial derivatives normal to the concatenation of two of these subdomains. The rest of operations are local for each subdomain. The solution of the tridiagonal equation system at all gridpoints from several subdomains is performed by the Thomas algorithm. This comprises three recursive loops which have to be computed at every intermediate Runge-Kutta step.

Sergio Pérez Roca

14


3

COMPUTATIONAL SETUP

3 Computational Setup 3.1 Baseow 3.1.1 Mathematical model For the initial ow conditions in this thesis we have selected the ansatz of a plane laminar jet by Schlichting, [2], who developed a self-similar model establishing a relationship between the two dimensions of the jet, the longitudinal (x) and the transversal (y). assumptions in this problem.

First of all, it is worth it to make clear the considered

A plane jet is emerging into a stationary environment of identical uid

from a two-dimensional aperture at

x = 0.

As the jet propagates at constant pressure and there exist no

restraining walls, it must have a constant momentum ux

Z

J

at any cross section (x

= const.):

+∞

u2 dy = const

J =ρ

(3.1)

−∞ equation which assumes zero-drag in a constant-pressure control volume. Schlichting (1933) demonstrated that on the condition that boundary-layer approximations are applicable, the jet entrainment propagates following the cube root of

x,

so the resulting stream function

ψ

is:

ψ = ν 1/2 x1/3 f (η) where

η

(3.2)

is the self-similarity variable:

η=

y

The agreeing speed components are:

f 0 (η) 3x1/3

(3.4)

−ν 1/2 (f − 2f 0 η) 3x2/3

(3.5)

u= v=

(3.3)

3ν 1/2 x2/3

This set of equations leads to a dierential equation problem of the following expression, based on the auxiliary function

f: f 000 + f f 00 + f 02 = 0

whose boundary conditions are symmetry along the uid medium (u

=0

at

y = ∞).

x

axis (v

(3.6)

=0

and

∂u/∂y = 0

at

y = 0,

and a still

Expressed in the similarity variables this is:

f (0) = f 00 (0) = 0

(3.7)

f 0 (∞) = 0

(3.8)

These sort of equations are usually solved numerically. However, Schlichting (1933) obtained the exact analytic solution, consisting in:

f (η) = 2a tanh aη Sergio Pérez Roca

15

(3.9)


3

COMPUTATIONAL SETUP

3.1

Baseow

f 0 (η) = 2a2 sech2 aη Therefore the jet speed prole follows the symmetrical The constant parameter

a

+∞

J =ρ −∞

2a2 sech2 aη 3x1/3

from which we attain:

a= sech 0 = 1,

sech2 y

form, similar to a Gaussian distribution.

is dened by evaluating the momentum ux

Z

As

(3.10)

9J √ 16 ρµ

2

3ν 1/2 x2/3 dη =

1/3 ≈ 0.8255

J

equation (3.1):

16 1/2 3 ρν a 9

(3.11)

J 1/3 (ρµ)1/6

(3.12)

the maximum value of velocity, found at the centreline, is:

umax

2a2 2 = 1/3 = 3 3x

9 16

What implies a centreline velocity drop with

2/3

x−1/3 .

J 2/3 ≈ 0.4543 (ρµx)1/3

J2 ρµx

1/3 (3.13)

The following denition consists in one of the basic

ones all over this thesis, since it represents the characteristic length considered in the Reynolds number (see section 3.1.2). This is the parameter

b,

the width of the jet, equivalent to twice the distance

y

where

u = 0.01umax : b = 2y |1% ≈ 21.8

x 2 µ2 Jρ

1/3

It is relevant to point out that these expressions are not valid for

(3.14)

x = 0, in which we obtain indeterminacy.

Moreover, boundary-layer approximations are not suitable if the Reynolds number is small, what results in invalidity of the solution in these cases, meaning ignorance of any details of the ow near the jet exit.

3.1.2 Reynolds number denition The expression for the Reynolds number

Reb

used all over this thesis is the following:

Reb =

ρ˜∞ umax b µ ˜∞

(3.15)

ρ˜∞ and µ ˜∞ are the density and the dynamic viscosity corresponding to quiescent air at International ˜∞ = 288.15K , p˜∞ = 1atm = 1.013 × 105 P a, ρ˜∞ = 1.225kg/m3 Standard Metric Conditions, that is T −5 and µ ˜∞ = 1.789 × 10 kg/(m · s). In order to be coherent at every calculation, these reference values

where

of the Reynolds number will always be regarded as constant, since we are not considering heat transfer phenomena and therefore temperature keeps constant; and what's more important, these will be the initial baseow values for the entire temperature, pressure, and density elds. The

umax ,

as dened above, is the maximum centreline velocity. We are going to x this parameter by

keeping the maximum Mach number of the problem (at the nozzle exit) constant and equal to 0.2, with the aim of maintaining the same compressible behaviour, avoiding possible distortion of the solution. Thus, it is considered as an incompressible ow. Then, with a heat capacity ratio

Sergio Pérez Roca

16

κ = 1.4

for the air, an


3

COMPUTATIONAL SETUP

R = 287,

3.1

and assuming a constant temperature equal to

T˜∞ ,

we obtain a value of

umax = 68.0525m/s.

The last parameter within this Reynolds number specication is the characteristic length tioned in the previous section,

1%

b

Baseow

b.

As we men-

represents the width of the jet, until the axial velocity decreases to a

y -coordinate.

of its centreline value, both for negative and positive

Since we have xed the rest of

parameters, there is only the jet width left to produce changes in the Reynolds number. Consequently, at higher Reynolds numbers, the jet will have a wider core at the beginning with considerable ow speeds, but always lower than the dened maximum one. Taking into account the Schlichting's equations and expressions exposed in section 3.1.1, the methodology carried out to perform changes in the Reynolds number is the following. We dene a desired Reynolds number for our simulation, for example

Reb = 300,

which is the minimum number regarded in this thesis.

For this Reynolds number, since the rest of parameters are xed, we obtain a

b = 6.4380 × 10−5 m.

Now,

in order to be coherent with the Schlichting's plane jet model, we have to check the dependencies of this parameter with the rest of variables in our problem. By evaluating the axial velocity (equation 3.4), corresponding to

0.01umax ,

u

at

y = b/2

and making use of the expressions 3.3, 3.10, 3.13, we can

deduce an equation with only one unknown: 1/3

2umax 3x0 2

sech

2

q

0.01umax =

1/2 2/3

3ν∞ x0

(3.16)

1/3

s 0.01 = sech2 

x-coordinate

b/2

3x0

which can be reduced to:

That unknown is the

1/3

umax 3x0 2

b2



umax  24x0 ν∞

which will be designated as

x ˜0 ,

(3.17)

the point from which our jet will be

considered to emerge in to the quiescent environment at an initial centreline velocity and a with characteristic length

b.

By attaining this value, the

a

and

J

umax = 68.0525m/s

parameters, characteristics of the

jet, are also determined. Their value is easy to obtain employing the equations 3.12, 3.13 or 3.14. It is relevant to point out that as the Reynolds number increases, so do the initial point is situated in a further position from the theoretical

b

and

x ˜0 ,

what implies that the

x = 0, and so in a more developed region

of the jet. This is a necessary consequence of the ansatz of keeping the Mach number constant, and only varying the characteristic length in accordance with the plane jet equations. The Reynolds number range studied spreads from

Reb = 300

until

Reb = 2500,

considering it a wide

enough interval to study instability generation and development. If we compare our Reynolds number approach with the literature, it is certain that we have adopted a slightly dierent ansatz, with the aim of preserving the Mach number. Usually the Reynolds number is dened regarding the nozzle diameter and the bulk mean velocity described in section 1.2, though the latter being approximate to the centreline value in smoothly contoured nozzles, as Ravinesh C. Deo did, [1]. Some authors, like Boersma

et al, [15],

only calculate with one Reynolds number, 2400, based on the nozzle diameter. It is true though that they study especially the inuence of the velocity inlet prole, rather than of the Reynolds number. Ravinesh C. Deo, [1], conducted experiments with Reynolds numbers ranging from 1500 to 16500, also based on

Sergio Pérez Roca

17


3

COMPUTATIONAL SETUP

3.1

Baseow

the nozzle diameter. These values clearly go beyond the scope of this thesis, remaining at low Reynolds numbers with the objective of studying perturbations. Other authors such as Stanley

et al, and

et al, [17], or Klein

[18], have chosen lower Reynolds numbers, closer to the ones in this thesis, being

ReD = [1000, 6000]

respectively.

ReD = 3000

The criterium for the formers was the intention of keeping the

dissipation caused by ltering to less than 1% of the viscous dissipation.

Babucke, [20], performed a

dierent Reynolds number ansatz, basing it upon the vorticity thickness as a characteristic length, since he was more interested in aeroacoustics.

3.1.3 Domain dimensions Our domain is a basically a cartesian grid. The axes of this domain have been non-dimensionalised with respect to the characteristic length of the case, the jet width

b, so as to obtain comparable data for several

Reynolds numbers. We have always kept the same dimensions of this domain all over the comparisons of dierent initial conditions. Nevertheless many simulations were carried out to obtain the optimum scale with which to visualise well the jet behaviour, both considering the observation of instability formation and development all over the

x-axis,

and the uid entrainment all over the

y -axis.

First of all simulations were performed with equal dimensions for both axes, concretely 10 widths or diameters, obtaining the following results. Two of the simulated axial velocity contours are presented in the Figure 3.1 below, for the cases of

Reb = 300

and

Reb = 1000.

(b)

Axial velocity contours for Reb = 1000,

10bx10b domain (a)


10bx10b domain

Figure 3.1:

Results for the 10bx10b domain

Then it was thought to increase the domain due to lack of development observation in the was chosen an

Reb = 300

and

x-span of 17.5b while the y -span Reb = 600 (Figure 3.2).

Sergio Pérez Roca

was kept constant.

18

x-axis.

It

Below the corresponding cases for


3

COMPUTATIONAL SETUP

3.1

Axial velocity contours for Reb = 300, 17.5bx10b domain

(b)

(a)

Baseow


17.5bx10b domain

Figure 3.2:

Results for the 17.5bx10b domain

Again, and although ner meshes were used, it was still complicated to study the development of instabilities in such a reduced space. Consequently, it was decided to increase considerably the size of the domain, more than four times the previous one,

85bx60b.

In this case it was nally possible to visualise in depth

the instability development and jet entrainment, observing that the jet inuences the quiescent ow a remarkable number of initial widths away. Below the corresponding cases for

Reb = 600

and

Reb = 1000

in Figure 3.3:

(a)


(b)

85bx60b domain

Figure 3.3:


85bx60b domain

Results for the 85bx60b domain

Making a comparison with literature, Boersma

et al,

[15], also agree with the idea that the domain

must be large enough in order to contain the entire ow eld. They attribute this requirement to the diculty of simulating free turbulent ow, where there is uncertainty in the election of boundaries and ow connement.

Hence, they took a length equal to 45 orice diameters, slighlty greater than the

half of our domain, what means that we can capture more downstream eects. Stanley Klein

et al, [18], conceived even smaller domains with 15Dx16D

Sergio Pérez Roca

19

and

20Dx8D

et al,

[17], and

respectively. The domain


3

COMPUTATIONAL SETUP

3.1

Baseow

dimensions of Babucke, [20], are not comparable since they are grounded on the vorticity thickness, reaching non-dimensional values of 300 in the longitudinal axis.

3.1.4 Mesh selection For the baseow mesh also several approaches have been carried out, but all of them with rectangle-shaped elements, since it was coherent with the shape of the domain, and there are no rigid bodies or surfaces considered. When the early small-domain calculations were performed, the employed strategy consisted in a higher element or point density all over the central region of the domain in terms of taking exactly half of the domain with center at equivalent to 1/4 of the

y -span,

y = 0.

y

coordinates,

The external parts, each one of them being

had an element length of 1.5 times the inner ones, since they are regions

less aected by the jet. Actually the programming method applied consists in dening some transition points, and establishing a fraction of the total number of gridpoints for each region, having the external ones 1/5 of the points each, and the central one the remaining 3/5. Thereby considerable computing time was saved.

There was no dierentiation made regarding the

x

coordinate, in view of jet development

accuracy objectives. Below there is an example of the mesh employed at the early calculations (Figure 3.4).

Figure 3.4:

Mesh for the 10x10 domain at Reb = 1200

The number of gridpoints at the early calculations was considerably low, being 120 in the in the

y -axis.

It was decided to include more points in the

characteristics in that direction. When the domain's

x

y -axis

x-axis

and 200

due to the quick evolution of the jet

length was increased to 17.5, the domain had to

be divided into 6 subdomains, with the aim of reducing computing times. Nevertheless the number of gridpoints was not a varied parameter. When me moved to a greater domain (85bx60b), a study of the necessary gridpoints was performed, looking at the stability of the solution.

Sergio Pérez Roca

In other words, an optimum number of gridpoints points was

20


3

COMPUTATIONAL SETUP

3.1

Baseow

found out, with which the computational resources could be the minimum needed to obtain an accurate solution of the problem. The method observed was simply empirical, checking the variation of the DNS result for a given Reynolds number (Reb

= 600)

until it kept constant from a given number of elements.

This optimum turned out to be, for a 24-subdomain approach, 200x275 points per subdomain, that is

1.32 × 106 points. The 24-subdomain distribution consists in 6 subdomain along the x axis and 4 along the y axis, with a numeration starting from 0 in the left lower corner, and increasing towards the right (greater x) and upwards (greater y ). In the next gures (3.5 1200x1100 in the overall domain, a total of

until 3.8) this studied trend can be seen.

Figure 3.5:

Sergio Pérez Roca


21


3

COMPUTATIONAL SETUP

Figure 3.6:


(a)

Baseow


Detail of axial velocity contours with 200x275 gridpoints per subdomain

Detail of pressure contours with 200x275 gridpoints per subdomain

(b)

Figure 3.7:

Sergio Pérez Roca

3.1

(c)

Density contours with 200x275 gridpoints per subdomain (d)


22


3

COMPUTATIONAL SETUP


(a)

3.1

Detail of axial velocity contours with 215x300 gridpoints per subdomain

Detail of pressure contours with 215x300 gridpoints per subdomain

(b)

Figure 3.8:

(c)

Baseow

Density contours with 215x300 gridpoints per subdomain (d)


As it can be observed, the ow instability shape and position are almost identical for the last two cases. Moreover the data set information gathered in

R Tecplot

is also coincident. Thus we concluded that a

200x275 gridpoints per subdomain conguration was accurate enough for our objectives. Furthermore it is important to point out that the element distribution strategy was slightly dierent. Seeing that the number of elements increased considerably with the bigger domain size, and thinking of the computational cost, the

y -span

was splitted in 5 parts. The greater of them was the central region, with a width of 1/3

the whole span and centred at

y = 0.

Then, the external parts consisted in 4 regions. 2 upper and lower,

with a width of 1/6 of the span each. The size of the elements increases inasmuch as we move outside the centreline. In the innermost external region, the element measures 10/3 times more than in the central region, and in the outermost it measures 1.5 times more than in the innermost external region.

This

strategy can also be expressed as follows. The outermost regions contain a 1/15 of the total points each, the second regions 1/10 of the points each, and the central the remaining 2/3. be seen in the following subgures within Figure 3.9.

Detail of the meshing strategy 1

(a)



(b)

Figure 3.9:

(c)

Meshing strategy

It is worth it to indicate that the mesh was calculated with the

Sergio Pérez Roca

Detail of the meshing strategy 4 (d)

23

R function meshgrid, MATLAB

having


3

COMPUTATIONAL SETUP

3.1

Baseow

previously dened point distributions along both axes. This way a matrix with all the gridpoints' positions is obtained, in absolute coordinates, as we will explain in the next section. Moreover, in the following Figure, 3.10, a contour plot of the 24-subdomain approach is presented.

Figure 3.10:

Contour plots of axial velocity for the 24 subdomains at Reb = 1800

Contrasting with the literature, Boersma

2.3 × 106

et al,

[15], designed a computational grid of 450x80x64 or

points, though in a round jet 3D conguration. That is the double of our nodes, but considering

that our problem remains two-dimensional, our mesh is quite ner. Besides, Stanley their simulation with also a 3D conguration of 390x390x130 or mesh. Klein

et al, [18], resolved with 360x128x512 or 2.34 × 107

1.98 × 107

et al, [17], performed

nodes, this being a highly ne

points, even ner. These two latter also

dealt with plane jets, but in 3D. Babucke, [20], chose a 2500x850

≈ 2.125 × 106

gridpoints conguration.

His accuracy is notably higher considering its characteristic length approach.

3.1.5 Fields denition The obtained meshgrid was used to calculate the relevant elds for the velocity eld, composed by its axial component ature

T

and pressure

p

u,

transversal

v

NS3D

and spanwise

number and consequently a

Sergio Pérez Roca

η

the density

ρ,

Temper-

elds. For the velocity eld, the mathematical model explained in section 3.1.1

was used. First of all it is necessary to calculate the beginning of our domain, self-similar variable

program. These are the

w;

b

parameter, using equation 3.17, having

umax

x ˜0 , for a given ν∞ constant.

and

Reynolds Then the

is computed for all the elements of the meshgrid matrix (expression 3.3). Now at

24


3

COMPUTATIONAL SETUP

3.1

Baseow

the beginning we are working with dimensional variables so as to obtain results coherent with boundary

a and J parameters, by means of the formulas 3.13 and J . Next the functions f and f 0 were evaluated at every point taking equations 3.9 and 3.10. With all this computed data is possible to get the u ˜ and v˜ values at every element, by means of expressions 3.4 and 3.5. The w ˜ , spanwise component, was equated to zero at

layer theory. It is also required to calculate the 3.12, rst calculating

a

for the given

x ˜0

and then

the whole eld since we are considering a two-dimensional domain. The streamlines can also be calculated using the expression 3.2, just for visualisation in

R. MATLAB

In order to proceed to the non-dimensionalisation required by the

NS3D program to work, we have applied

the following conceptions. The velocity components are normalised regarding the maximum speed

umax .

The density and temperature elds are equal to one in all the eld, having been normalised over their free stream values, as this way:

NS3D

pnon−dim = p =

recommends. The pressure eld, also as

ρT , following the κM a2

over the characteristic length

b,

NS3D

NS3D

indicates, is normalised in

notation. Finally, the meshgrid is also normalised

as we mentioned before in the domain denition.

In gures 3.11 until 3.13 we can see here some pictures of the baseow calculated with

R introduced MATLAB

in the CFD programme. In Figure 3.11 the velocity eld is represented, in Figure 3.12 the velocity contours and in Figure 3.13 the employed mesh, for a given

Figure 3.11:

Sergio Pérez Roca

Reb = 600.

Velocity eld for Reb = 600

25


3

COMPUTATIONAL SETUP

Figure 3.12:

3.1

Baseow

Axial (dimensional and non-dimensional) and transversal velocity contours for Reb = 600

Figure 3.13:

The edition of the baseow les with

Sergio Pérez Roca

.eas

Mesh for Reb = 600

format is performed by means of a code provided by the

26

IAG


3

COMPUTATIONAL SETUP

3.1

Baseow

member Björn Selent. Thereby 24 les of 2.5 MB each are obtained. As the rst simulations were performed and analysed, it was decided to include a little perturbation in this baseow elds so as to promote the earlier formation of jet instabilities, and save calculation resources. This perturbation consisted in adding a random number between 0.01 to the velocity components. It was implemented within the

−1 and 1 multiplied by an amplitude of

NS3D 's general code le ns3d_grid.f90.

Nevertheless this approach was rejected in the end due to unexpected interactions with the damping zone, explained in section 3.2.2. In the literature it is common to nd dierent approaches to the initial velocity eld denition. instance given hyperbolic proles at the inow as in the DNS of Stanley

For

et al, [17], enforced and provided

with energy by a dened three-dimensional energy spectrum. Another ansatz was carried out by Boersma

et al, [15], who experimented with not only top hat velocity proles but also rather articial ones. Klein et al, [18], also used hyperbolic-tangent proles with sumperimposed uctuations to generate a turbulent inow. Babucke, [20], based his baseow elds upon boundary layer theory, considering self-similarity for

the mixing layer of the jet and also the Blasius boundary layer. That is the closest strategy to ours found. One of our objectives is checking the validity of Schlichting's solution, [2], so we will keep to this baseow approach to see its evolution.

3.1.6

(Graphical User Interface) for the creation of baseow les with R LAB GUI

MAT-

With the aim of making the process of looking for the optimum baseow les as quick and intuitive as possible, a

GUI

(Graphical User Interface) was programmed in

R. MATLAB

Due to the considerable

amount of baseow parameters involved in the search of a suitable frame of visualisation of the plane jet, this tool was of great use. It allows the variation of the Reynolds number with a resolution of 100, the Mach number, if other initial velocities are to be checked; the domain dimensions, number of subdomains and amount of nodes per subdomain in both directions. It contains also a button which plots the velocity eld, contours of velocity and mesh in separate gures; as well as another button initiating the writing of the baseow les when all the parameters have been selected. Two naming options for these les are oered, one including the Reynolds number and the other just following the necessary designation for the

NS3D

simulation. Moreover, the

ns3d.i

le parameters, commented on in section 3.2.3 can also be

obtained by introducing some required data, those being the security factor of the simulation, the number of jet periods desired and the optimum product of fundamental frequency and time steps period, also dealt with in section 3.2.3. In Figure 3.14 there is a sample of the interface.

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

Figure 3.14:

3.2

R Sample of the MATLAB

GUI

Calculation parameters

for Reb = 600

3.2 Calculation parameters 3.2.1 Boundary conditions Having our baseow ready, the next step to explain are the selected boundary conditions.

It was also

necessary to perform several calculations in order to check the validity of these basic conditions in a CFD code. Since our domain is relatively simple in terms of boundaries, the inow and the outow constraints did not pose relevant problems, obtaining positive results in the early tries. The inow condition is dened by a zero pressure gradient in the

x

axis with extrapolation of pressure,

as typical in the jet problems, since it is not a pressure driven ow, but a momentum driven one. This condition corresponds in the

NS3D

code to a

-5.

The velocity and temperature eld are kept constant

throughout the calculation, preserving the baseow values.

This is also logical since we consider the

jet exit as steady. Since the pressure is extrapolated and the temperature is xed, the density is easily calculated by means of the ideal gas law. The ow in the lower

x-coordinate region did not seemed disturbed by this constraint, that is to say, there

were neither recirculation of streamlines nor unexpected gradients observed. In this type of problems it would not be desirable to obtain ow coming out at the inow boundary inasmuch as the whole approach would not make sense. The highest streamwise velocities are found here. Thus, it was kept as a valid one, as it can be seen in Figure 3.15.

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3

COMPUTATIONAL SETUP

Figure 3.15:

3.2


Inow boundary condition detail with axial velocity contours and streamlines for Reb = 1000

The outow restraint was rst checked to work as an extrapolation of all the eld variables, but due to instabilities in the programme, it nally consisted in a zero-gradient one, dening all the eld variables as equal as in the previous

x

coordinate. Moreover the time derivatives of the elds are equated. This

conception leads to a smooth exit of the ow out of the control volume, avoiding undesirable gradients, which may distort the upstream ow. It corresponds to a

-14

in the

NS3D

code, with the following basic

relationship:

∂Q ∂Q |Nx = |N −1 , ∂t ∂t x Q

being any of the elds, and

Nx

the total number of nodes in the

(3.18)

x

axis.

It was checked to be suitable, considering that it should also be able to allow ow with opposite direction, caused by instabilities or eddies. Furthermore it is only valid for subsonic ows, which is the case. There is below Figure 3.16 as a proof.

Figure 3.16:

Outlow boundary condition detail with axial velocity contours and streamlines for Reb = 800

The constraints posing more problems turned out to be the lateral ones, rst considering inow and outow conditions, taking into account the entrainment from both sides, but rapidly led to calculation crashes with oating points exceptions. Therefore another conception was adopted, a free stream condition. In the programme there are several options to dene the parameters of this free stream. First the characteristic free strom condition (

-21 )

was used, with the features that it cancels the in-owing perturbations but

extrapolates the out-going ones, and the addition of the characteristic variables to the baseow. It was also implemented a smoothing region within. Nevertheless it resulted in too steep gradients and too much interaction with the jet, obtaining strange contour distribution and too forced streamlines, as can be checked in Figure 3.17.

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COMPUTATIONAL SETUP

3.2


Characteristic free stream boundary condition detail with axial velocity contours and streamlines for few gridpoints (a)

Characteristic free stream boundary condition detail with axial velocity contours and streamlines for more gridpoints (b)

Characteristic free stream boundary condition for Reb = 600

Figure 3.17:

Therefore another type of free stream condition, the simplest one corresponding to This implies zero gradients in

y

at the limit are calculated with the following 2

Φj=my = where

Φ

-20,

was analysed.

axis all over the limits. The values for velocity, pressure and temperature

nd

order partial dierentiation.

4Φj=my−1 − Φj=my−2 , 3

(3.19)

is any of the elds. Subsequently pressure is calculated thanks to the ideal gas law.

This will dump eectively all the undesirable steep gradients and allow a simple entrainment. Figure 3.18 shows these results.

Figure 3.18:

Free stream boundary condition for Reb = 600

Having dened the boundary conditions, it is relevant to point out that calculating with a 24-domain conguration, the setup of the

NS3D's gebiet.cong

le, which serves as a specication of the connection

between subdomains and its boundary conditions, had to be reprogrammed adequately, taking into account the aforementioned numeration of subdomains (subsubsection 3.1.4).

Sergio Pérez Roca

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COMPUTATIONAL SETUP

3.2

Taking a look at the literature, Boersma

et al,


[15], were especially comprehensive on the selection of

these boundary condition, with the main aim of allowing full transparency fot the entrainment ow at the lateral boundaries. For these boundaries they employed a so-called traction-free condition, allowing velocity across the boundary and therefore entrainment. They observed that this type of constraint resulted in the same streamline pattern as Schlichting (1933). The lateral condition adopted in this thesis also permits the velocity across itself though with no gradient, allowing entrainment. At the inow they specied the velocity components and left the pressure free, and at the outow they chose the so-called convective boundary condition, thinking of stability. Besides, in the DNS of Stanley

et al, [17], the inow boundary was considered as subsonic characteristic,

and the outow and sidewall boundaries as the non-reecting Thompson (1987, 1990) condition.

The

mode of these conditions is permitted to change between that for non-reecting inow and outlfow at each node of the limit with dependence on the instantaneous local normal velocity. Klein

et al, [18], imposed zero-velocity outside the nozzle and a hyperbolic prole for the velocity at the

nozzle for the inow boundary, Neumann conditions for outow and set pressures to zero at the lateral limits interpolating tangential velocities, what allows mass entrainment. Furthermore, Babucke, [20], used both subsonic and supersonic boundary conditions at inow and outow, the formers being forced with eigenfunctions from linear stability theory. At the lateral boundaries he used the characteristic freestream condition.

3.2.2 Damping zone or

sponge

Another tool used while simulating was the damping zone or

sponge

at the end of the domain. In the

damping zone, force terms counteract deviations from the baseow in the governing equations. Strictly speaking, the damping zone is therefore not a constraint but a volume force of the edges. Hence, it can be combined with all other boundary conditions. The gain of these force terms is weighted by means of a dened 5

th

order polynomial at every boundary specied. This polynomial serves as a transition function

between two domains, in this case from a not damped to a damped region ended by an outow boundary layer.

Its fth grade stems from the condition that the rst and second derivatives at the boundaries

must be zero. The resulting form of this function is:

σ(x∗ ) = 1 − 6 · x∗5 + 15 · x∗4 − 10 · x∗3 with

x∗

(3.20)

being the non-dimensional range of the transition zone:

x∗ =

x − x0 ∆xramp

(3.21)

While analysing results in terms of whole values of non-dimensional axial velocity, the damping zone tool did not seem necessary. However, when we started to look at the axial velocity perturbation part, the need for this attenuating eect was manifest, as we will explain in the following paragraphs.

−2 or

Since the perturbations are of considerably lower order than the maximum axial velocity (10

10−6

depending on the type of calculation explained in section 3.2.3), the eect of the end boundary condition

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2


is stronger, aecting the entire domain. In Subgures 3.19, in which a simulation without

sponge

and

with the rst approach of it are compared, this severe impact is obvious.

Contours of axial velocity perturbation without sponge by normal calculation at Reb =

Contours of axial velocity perturbation with sponge of gain 3 at the nal 5% points of the domain by normal calculation at Reb = 600

(a)

(b)

600

Figure 3.19:

Comparison between no-sponge and

sponge

approaches of gain 3 last 5% by normal calculation at

Reb = 600 As we can see, the eect of the end boundary at Subgure 3.19a is the creation of great bubbles of perturbed zones which are sically senseless. Nevertheless, with a damping zone of gain 3 at the last 5% of the points of the domain (Subgure 3.19b), which was the rst approach, the domain is also highly disturbed, amplicating the perturbations upstream and having a hard damping eect at the end, making disturbances at the centreline disappear. Consequently the gain of this zone was reduced by a factor of 6, with the aim of having a smoother attenuation.

In Figure 3.20 we can see the result for a normal

calculation of gain of 0.5.

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2


Contours of axial velocity perturbation with sponge of gain 0.5 at the nal 5% points of the domain by normal calculation at Reb = 600

Figure 3.20:

Again we see that even with this small gain the upstream ow is also highly disturbed, being a peculiar and unexpected result, though the attenuation at the outow seems satisfactory. After reecting upon it and making some tests, it was checked that this eect was due to the interaction between the damping zone and the random perturbation included in the baseow, in order to promote instabilities.

In the

following gures these results are shown, Figure 3.21 refering to a normal calculation and Figure 3.22 to a disturbance calculation, which was also tested, as explained in section 3.2.3.

Contours of axial velocity uctuation without random perturbation, with normal calculation at Reb = 600

Figure 3.21:

Sergio Pérez Roca

33

sponge

of gain 0.5 by


3

COMPUTATIONAL SETUP

3.2

Contours of axial velocity uctuation with random perturbation, with sponge of gain 0.5 by disturbance calculation at Reb = 600


Contours of axial velocity uctuation without random perturbation, with sponge of gain 0.5 by disturbance calculation at Reb = 600

(a)

(b)

Comparison between randomly perturbed and not perturbed approaches of gain 0.5 by disturbance calculation at Reb = 600 Figure 3.22:

It is also interesting to show the nal evolution from a no-

sponge

and a

sponge

approach for both normal

and disturbance calculations, in Figures 3.23 and 3.24.

Contours of axial velocity uctuation with random perturbation, without sponge by normal calculation at Reb = 600

Contours of axial velocity uctuation without random perturbation, with sponge of gain 0.5 by normal calculation at Reb = 600

(a)

Figure 3.23:

(b)

Comparison between no-sponge and sponge approaches of gain 0.5 by normal calculation at Reb = 600

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2

Contours of axial velocity uctuation with random perturbation, without sponge by disturbance calculation at Reb = 600

Contours of axial velocity uctuation without random perturbation, with sponge of gain 0.5 by disturbance calculation at Reb = 600

(a)

Figure 3.24:


(b)

Comparison between no-sponge and

sponge

approaches of gain 0.5 by disturbance calculation at

Reb = 600 It is observable that the random perturbation is not easily noticeable when the

sponge

is not used, which

proves that the interaction between the damping zone and the random perturbation is an important phenomenon, relevant to be studied in future projects.

3.2.3

NS3D's ns3d.i

le parameters

One of the basic les to edit when simulating with

NS3D

is

ns3d.i

le, serving as a means of specifying

M a, the Prandtl number Reb , the latter

fundamental parameters of the simulation. On the one hand physical, the Mach number number

P r,

the reference temperature

T˜∞

and pressure

p˜∞ ,

and the Reynolds

being varied and the rest kept constant. As the uid in the domain is air, the Prandtl number was dened as 0.71, as told in section 2.3. On the other hand, the temporal parameters of the simulations have also to be selected, those being the

ω0 , the period of the time steps ZS _period, the number of time steps included in the ZS _output, the starting time step of the calculation ZS _start, and the desirable last one

natural frequency output les

ZS _ende. The number of time steps taken for the output les has always been 50 in order to obtain comparable frames of study, important for post-processing. The rest of parameters are dependent on the Reynolds number and baseow characteristics.

The start time step was logically established at 1 for every new

calculation, except when they were performed in several phases. The last time step was dened so as to reach a jet development phase after transition to non-linearity, experimentally checked at each Reynolds number and explained in section 4.1.1, but from a minimum of around 2.8 jet periods.

A natural jet

period expresses the non-dimensional time needed for the passage of a uid particle from the centreline

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2


since it is exhausted in the nozzle until it leaves the domain, at initial conditions. This was calculated by obtaining the mean value of the non-dimensional axial velocity at the centreline and relating it to the non-dimensional

x

also checked with

length of the domain. That mean value was easily determined with

R after Tecplot

R , but MATLAB

the initial short test simulations for each Reynolds number, as showed

in Figure 3.25, obtaining similar results.

Figure 3.25:

R at Reb = 2500 Mean axial velocity at centreline obtained with Tecplot

Nevertheless, in order to dene the last time step regarding the natural jet periods, it is necessary to know the appropriate time step length, which is determined by the natural frequency and the period of time steps. An optimum selection of these parameters has to be made for every dierent initial conditions in order to obtain a CFL security factor (Courant-Friedrichs-Lewy condition) close to 1. This factor is dened by:

CF L =

c∆t , ∆x

(3.22)

establishing a relationship between the time and spatial step length and the convective velocity of the problem (c). Since the spatial step length is determined by the optimum mesh and the convective velocity is a value calculated by the

NS3D, not directly available to the user, the convergence had to be carried out

empirically. That is another of the reasons why initial short test simulations are required while conguring new cases. Knowing the CFL value for the test, the product of the natural frequency and the period of time steps is adapted, seeing that they are both directly proportional, the latter representing the time step length. Once knowing the time step length, easily calculated by means of:

∆t =

2π , ω0 . ZS _period

(3.23)

we can get the number of time steps necessary to reach the desired number of natural jet periods. Then, the period of time steps must be a divisor of the total amount of time steps calculated. Therefore, it is simply equated to it, in view of simulation reproducibility. After this step, the natural frequency can be easily determined having in mind the previous product found out.

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2


Disturbance calculation application In this le it is also possible to dene the type of calculation being carried out. Since the beginning of this project and until the start of the velocity uctuation study standard DNS calculations have been performed. Nevertheless, while looking at the early perturbation results, it was thought to change the approach to a disturbance calculation, since the uctuation distributions turned out to be of relatively high orders, almost of the order of the maximum velocity at the nal periods, and presented chaotic shapes from quite early time steps, not being suitable for the subsequent instability postprocessing. Moreover, disturbance calculation is more coherent with our objectives related to the initial self-similar solution, inasmuch as this ansatz adopts the given baseow as a steady-state solution of the Navier-Stokes equations (section 2.4.2). Therefore we are also checking the deviations of the developed ow from the proposed solution. This arguments can be proved in the following gures, in which normal and disturbance calculations are compared at several time steps, always without the random perturbation used before, as it has been proved to be problematic in the previous section (3.2.2). The label one and

stoer_calc

normal_calc corresponds to the normal

to the disturbance one. In Figure 3.26 they are compared at a short time after the

beginning of the simulation, at 0.05 jet periods.

Contours of axial velocity uctuation by normal calculation after 0.05 jet periods at

Contours of axial velocity uctuation by disturbance calculation after 0.05 jet periods at Reb = 600

(a)

(b)

Reb = 600

Comparison between normal and disturbance calculation for axial velocity uctuation after 0.05 jet periods at Reb = 600 Figure 3.26:

It is observable that even at that short time after the start the regular simulation presents a region of small uctuations all over the jet developing area, whereas the disturbance does not present any, the values being almost identical to the baseow ones. In Figure 3.27 they are compared at 0.5 jet periods, when the perturbations have been seen to start at more or less the same

−4 for the normal and with totally dierent orders of magnitude, 10

Sergio Pérez Roca

37

x coordinates for both cases,

but

10−10 for the disturbance one.


3

COMPUTATIONAL SETUP

3.2

(a)


(b)

Reb = 600

Reb = 600


Contours of axial velocity uctuation by disturbance calculation after 0.5 jet periods at


In Figure 3.28 the comparison is made at 1 jet period, with the outcome that the perturbed wave rises more in the disturbance calculation, to an order of

10−8 ,

and the normal one only rises one order of

−3 ), but the latter propagates slighlty faster towards the outow. magnitude (to 10

(a)

Contours of axial velocity uctuation by normal calculation after 1 jet period at Reb =

(b)

Contours of axial velocity uctuation by disturbance calculation after 1 jet period at

600

Reb = 600

Comparison between normal and disturbance calculation for axial velocity uctuation after 1 jet period at Reb = 600 Figure 3.28:

In Figure 3.29 the jet periods are 1.6, resulting that the perturbed wave rises almost two orders of

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2


magnitude in the disturbance calculation, and the normal rises approximately the same, with the dierence that in the normal one the second perturbed wave has greater relative amplitudes to the rst wave than in the disturbance one.

(a)


(b)


Reb = 600

Reb = 600


Figure 3.30 shows the comparison after 2.2 jet periods, in which it is clear that the normal calculation breaks down into non-linearity considerably earlier than the disturbance one.

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

3.2

(a)


(b)

Reb = 600

Reb = 600




Figure 3.31 illustrates the comparison after the nal time steps calculated in the rst ulations, that is 2.8 jet periods.

Reb = 600

sim-

Here it is discernible that the amplitudes of the normal calculation

perturbations are one order of magnitude higher than the disturbance one (10

−2 to

10−3 ),

and that

non-linearity is much more developed in the former, having such chaotic distributions.

(a)


(b)


Reb = 600

Reb = 600


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COMPUTATIONAL SETUP

3.2


In conclusion, the disturbance calculation was seen as a more suitable approach regarding our objectives, and therefore they were broadened to several Reynolds numbers. The main disadvantage is the need for more time steps in order to visualise the entire evolution of the perturbation and the start of transition into non-linearity. But on the other hand more resolution in time is gained. Below there is an example of the denitive le edition (Figure 3.32).

Figure 3.32: ns3d.i

3.2.4

NS3D's setup

With regards to the

setup

edition sample at Reb = 2500

le parameters le, it is worth it to comment on the computational resources employed. Since

a 24-subdomain ansatz is being used and it remains as a 2D simulation, 24

MPI

(Message Passing Inter-

face) processor boards, one for each subdomain, have been available for parallelisation in order to enhance the simulation speed. Each of the boards could be used with its maximum calculation power, that is 32

OpenMP

(Multi-Processing) threads. With this calculation resources, simulations could be performed in

around 17 hours per jet period at the lowest Reynolds number, and around 3,5 at the highest, since the number of time steps necessary for low steps per period for

Reb = 300

Reb

to 53000 for

cases is considerably higher. These range from 276000 time

Reb = 2500.

As more than one period was needed to study

transition, computing times reached 5 days at the lowest Reynolds number case and 9,5 hours at the greatest. Finally it is also important to dene in this le how frequently the programme has to save a copy of the calculated iterations in terms of amount of time steps.

Sergio Pérez Roca

41

This was kept to 1000 due our relatively high


3

COMPUTATIONAL SETUP

3.2


amount of total time steps. Below there is also an example of the le edition (Figure 3.33).

Figure 3.33: setup

edition sample at Reb = 2500

3.2.5 Three-dimensional test After dealing with rst two-dimensional calculations, it was also decided to check the viability of a threedimensional calculation, conceding the plane jet a spanwise development or width. It was implemented by dint of the spectral method, allowing a periodic solution in the

z -axis,

suitable boundary condition for

a plane jet. With more restrictive boundaries, such as walls, the ow would have been considerably more disturbed. Calculations were performed with a with of 10 spanwise modes, and the eld was divided in 6 subdomains. Nevertheless, this type of calculations were rejected due to the need for introducing high initial random perturbation amplitudes in order to trigger instabilities in the parallel for all

z

z -axis.

Otherwise the ow was completely

planes and the third dimension was senseless, there was practically no entrainment in the

spanwise direction. And on the other hand, with those high amplitudes our initial self-similar solution was exceedingly distorted. In the following Figure, 3.34, the results for a 3D calculation with included random perturbations of amplitudes one order of magnitude lower than the proper velocity eld of the self-similar solution are shown. It can be seen that the ow is no longer parallel in the

z -axis

but it is

also extremely dicult to interpret. Consequently for the rest of the project only 2D simulations were contemplated, considering 3D ones meaningless for our purposes.

Sergio Pérez Roca

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3

COMPUTATIONAL SETUP

(a) Figure 3.34:

3.2

Spanwise view


(b)

Axial view

Contour plots of axial velocity with streamlines and isosurfaces for the 3D approach at Reb = 500

3.2.6 Computational setup summary After having explained the selected computational setup in detail, it is worth it to present a summary table so as to have everything in mind and facilitate the work for possible students who may take this thesis up again and continue with further postprocessing. Table 3.1 serves as so.

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COMPUTATIONAL SETUP

3.2

Computational setup of the common frame with

Reb = b

ρ˜∞ umax b µ ˜∞


NS3D

[300, 600, 800, 1000, 1200, 1800, 2500] [6.44 × 10−5 , 1.288 × 10−4 , 1.717 × 10−4 , 2.146 × 10−4 , 2.575 × 10−4 , 3.863 × 10−4 , 5.365 × 10−4 ](m)

M a = √umax˜

0.2

Pr =

0.71

κRT∞ c˜p µ ˜∞ ϑ˜∞

Freestream reference values

International Standard Metric Conditions

Baseow

Schlichting's self-similar model from his Boundary Layer Theory (1933)

Domain dimensions

85bx60b

Number of subdomains

24 in 2D (6x4)

Gridpoints

200x275 per subdomain, 1200x1100 overall what means

1.32 × 106

stoer_calc )

Type of calculation

Disturbance DNS (

Boundary conditions

Inow: zero pressure gradient (-5); outow: zero-gradient (-14); lateral: freestream (-20)

Damping zone

Gain 0.5 at last 5%

Spatial Discretisation

Compact Finite Dierences of 6

Table 3.1:

Sergio Pérez Roca

th

order (21)

Summary of the computational setup of the common frame

44


4

POSTPROCESSING OF RESULTS

4 Postprocessing of results In this section we are going to present the results of the simulations for the following Reynolds numbers and analyse them. As introduced before, the calculated Reynolds numbers chosen are

Reb = 800, Reb = 1000, Reb = 1200, Reb = 1800

and

Reb = 2500.

Reb = 300, Reb = 600,

Results in terms of velocity

perturbation are the main core of this postprocessing, but we will start by showing the base-state axial velocity, in order to acquire a general understanding of the processes that take place in this ow. For this task, the programmes

R , EAS3 Tecplot

and a spreadsheet have been used.

4.1 Axial velocity 4.1.1 Axial velocity elds Linear scale In Figures 4.1 to 4.7 the contour plots of axial velocity are represented, with a linear scale from -0.2 to 1 for Reynolds numbers from 300 to 800, and from -0.4 to 1 for higher, since 1 corresponds to the maximum inlet velocity, and the greater the Reynolds number the lower the minimum negative value of axial velocity. They are plotted at key time steps, selected so as to show the generation of perceptible instability waves and two-dimensional transition to non-linearity.

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4


4.1

Axial velocity

At start

(b)

After 4.75 jet periods

(c)


(d)


(e)


(a)

Figure 4.1:

Sergio Pérez Roca

(f )

After 7 jet periods

Contour plots of axial velocity at dierent time steps, at Reb = 300

46


4


4.1

Axial velocity

At start

(b)


(c)


(d)


(e)


(f )


(a)

Figure 4.2:

Sergio Pérez Roca


47


4


(a)

4.1

At start

(b)

Axial velocity

After 2 jet periods

(c)


(d)


(e)


(f )


Figure 4.3:

Sergio Pérez Roca


48


4


4.1

Axial velocity

At start

(b)


(c)


(d)


(e)


(f )


(a)

Figure 4.4:

Sergio Pérez Roca

Contour plots of axial velocity at dierent time steps, at Reb = 1000 49


4


4.1

Axial velocity

At start

(b)


(c)


(d)


(e)


(f )


(a)

Figure 4.5:

Sergio Pérez Roca



4


4.1

Axial velocity

At start

(b)


(c)


(d)


(e)


(f )


(a)

Figure 4.6:

Sergio Pérez Roca



4


4.1

Axial velocity

At start

(b)


(c)


(d)


(e)


(f )


(a)

Figure 4.7:

Sergio Pérez Roca



4


4.1

Axial velocity

In the previous gures we can see a clear dependence on the Reynolds number in terms of intensity of instabilities (amplitudes) and transition. Every simulation starts though with an apparent stationary phase with no deviation from the self-similar baseow solution, the longer the lower the Reynolds number. As we will comment on the subsequent section 4.2.1, there are mainly two linear sinusoidal instability wave packets that can be perceived in the whole axial velocity eld, but perceived in more detail in the perturbation eld. The amplitudes of these waves at start is some orders of magnitude lower than the jet core velocity. Consequently in the axial velocity contours stationary phases seem to occur, but they are not completely so. These waves start to be perceptible in the velocity contour earlier the greater the Reynolds number, which means that their increase in amplitude is more rapid and to a greater extent. The immediate consequence of a greater Reynolds number is then the generation of intense waves closer to the nozzle exit and so at earlier time steps. Moreover, the wavenumber is also higher, since more wave periods are present when the Reynolds number increases. Regarding the phenomenon always named as transition in this thesis, some clarication needs to be made. Since our problem is two-dimensional, turbulence cannot be dealth with, as it is a three-dimensional phenomenon by denition. Therefore, when we refer to transition it is always meant in the sense that the jet develops any longer linearly and starts its non-linear phase. In the light of the results, it is given at an earlier period with a Reynolds number increase, and it is characterised by the disturbance of the linear sinusoidal waves, and formation of non-linearity. Due to the more intense uctuation there are parts of the domain with negative values of axial velocity, the lower the Reynolds number. The following Table 4.1 summarises the observed time steps in which transition is observed to occur.

Table 4.1:

Reb

Transition [jet periods]

300

5.75

600

3.25

800

2.25

1000

1.9

1200

1.75

1800

1.15

2500

0.9

Transition in jet periods at each Reynolds number observed in velocity contours

Exponential scale Further velocity eld contours are presented in Figures 4.8 to 4.14 with an exponential scale, so as to see the interaction of the jet with the ambient uid. In order to comply with this scale, the contour variable needs now to be the absolute value of axial velocity. The same time steps as in linear scale are plotted. However, as start is not really interesting, the rst subgure of each gure represents instead the rst perception of instabilities in the jet surface. Depending on the time step, a dierent range is plotted, in

−15 to 1) has been

order to enhance visualisation of structures. At the rst subgures, the largest span (10

Sergio Pérez Roca

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4


4.1

Axial velocity

chosen, in order to appreciate the smallest disturbances. Subgures b to e from each gure have a range between

10−9

to 1, since jet development triggers higher velocities in the whole eld. Finally, subgures

f have a span from

Sergio Pérez Roca

10−6

to 1, since non-linearity is already present and the ow is higly disturbed.

54


4


4.1

Axial velocity

(a)


(b)


(c)


(d)


(e)


Figure 4.8:

Sergio Pérez Roca

(f )

After 7 jet periods

Contour plots of axial velocity at dierent time steps in exponential scale, at Reb = 300 55


4


4.1

Axial velocity

(a)


(b)


(c)


(d)


(e)


(f )


Figure 4.9:

Sergio Pérez Roca

Contour plots of axial velocity at dierent time steps in exponential scale, at Reb = 600

56


4


(a)


(c)


(e)


Figure 4.10:

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4.1

(b)

Axial velocity

After 2 jet periods

(d)


(f )




4


4.1

Axial velocity

(a)


(b)


(c)


(d)


(e)


(f )


Figure 4.11:

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4


4.1

Axial velocity

(a)


(b)


(c)


(d)


(e)


(f )


Figure 4.12:

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4


4.1

Axial velocity

(a)


(b)


(c)


(d)


(e)


(f )


Figure 4.13:

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4


4.1

Axial velocity

(a)


(b)


(c)


(d)


(e)


(f )


Figure 4.14:

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4


4.2


With this exponential scale we can corroborate the quicker and more intense formation of instabilities with a Reynolds number increase. Again the earlier and greater instability lobes can be seen, as well as the higher wavenumber. It is also patent that the whole domain is aected by instabilities at the jet, leading to an end result in which the order of magnitude of velocity is of order 0.1 or 1 in almost the entire eld. With these gures an idea of the processes involved in a jet development can be acquired, but not an exact assessment of wave packets. Therefore in the next section we will move on to analyse the perturbation component of velocity. Up to this point, it can only be stated that jet development and entrainment of environment uid are related, implying several phases of instability generation at the nozzle exit or start of the jet and amplication throughout the jet core, reaching transition to non-linearity.

4.2 Axial velocity perturbation 4.2.1 Axial velocity perturbation elds In this section several gures are displayed, concerning the longitudinal uctuation elds at dierent time steps for each Reynolds number. It is relevant to state, that these elds have been obtained by substracting the baseow velocity values, our self-similar steady solution, to the DNS solution at every time step calculated.

This operation has been carried out by means of the

component of velocity dimension of the jet,

v.

u

IAG 's

programme

EAS3.

The axial

has been chosen for the disturbance study due to its dominance over the other

Therefore, variations in

u (u0 )

are more characteristic and determinant than in

v.

The scale is exponential so as to present a comprehensive visualisation, since the orders of magnitude of perturbations increase rapidly, ranging from values around

10−15

at the beginning of the simulation

until reaching values of the order of magnitude of the base-state velocity. As the scale is exponential, the absolute value of the amplitudes is plotted. The chosen time steps correspond to 0.5, 1, 1.5, 2 and 2.5 jet periods. Nevertheless, the

Reb = 300

Reb = 600 cases are that Reb = 300 starts

and

time. It is not until almost 6 jet periods

special, due to their longer development its transition to non-linear regime, while

for the other cases transition occurs quite earlier, as seen before and as we will delve in this section. Therefore, Figure 4.17 includes time steps until 7 jet periods.

Reb = 600

does it around slightly later

than 3 periods. That way, Figure 4.19 includes time steps until 4 jet periods. All in all, Figures 4.17 to 4.24 show the contour plots of the absolute value of axial velocity uctuation at dierent time steps, one for each Reynolds number, varying from the smallest to the greatest.

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4.2


(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e)


(f )

After 3 jet periods

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4


4.2


(a)

After 3.5 jet period

(b)

After 4 jet periods

(c)


(d)

After 5 jet periods

(e)


(f )

After 6 jet periods

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4


(a) Figure 4.17:

4.2


(b)


After 7 jet periods

Contour plots of the absolute value of axial velocity uctuation at dierent time steps, at Reb = 300

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4


4.2


(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e)


(f )

After 3 jet periods

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4


(a) Figure 4.19:

4.2


(b)


After 4 jet periods


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67


4


4.2

(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e) Figure 4.20:




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4


4.2

(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e) Figure 4.21:




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4


4.2

(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e) Figure 4.22:




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4


4.2

(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e) Figure 4.23:




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4


4.2

(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e) Figure 4.24:




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4


4.2


From these gures the following trends can be seen. The formation of dierent wave packets of perturbation at approximately the same jet period steps and positions is manifest for all the Reynolds numbers. This is an interesting result, since it can be drawn that the generation of these waves depends on the relative development of each jet, in terms of the time needed for a particle to cross the domain.

Jets

with higher Reynolds number obviously develop faster, but the genesis of its instability packages always happens at the same phase of this development. The main cause for the generation of the rst wave is without any doubt the adaptation of the approximated solution or initial guess of the ow eld that we have introduced to the whole Navier-Stokes equations that the DNS code starts to solve when the simulation begins. With the self-similar model we are proposing some concrete elds to be a stationary solution of those equations, and so considering the temporal derivatives to be zero.

Nonetheless, the model is obviously not an exact solution of the N-S

equations, since some terms have been neglected so as to get the boundary layer equations, by considering it an area of dominant viscous interaction. Therefore, the numerical resolution of the whole equations from the early time steps has the immediate consequence of correcting the baseow, creating a perturbation that travels with the jet. The reason for the successive waves may be acoustic reexion in the end boundary condition, but we will assess this phenomenon in depth in section 4.2.3. An important similarity between the cases from

Reb = 300,

Reb = 600

to

Reb = 2500,

that is to say, all except for

is the clear distinction of two wave packets before transition, one emerging almost immedi-

ately after the beginning of the simulation, and the other generally after around 1.5 periods. In contrast, the wave responsible for transition is not the same, for Reynolds numbers until 1000 the second one is responsible, and from

Reb = 1200

it is the rst one. But for all of them the second wave achieves greater

amplitudes and gradients. The noteworthiness of the

Reb = 300

case is the appearence of more waves before its transition, and not

as cleary separate as with higher Reynolds numbers. The total number of waves originated is four, and the last two appear at the tail of the preceding wave. The third wave is generated around 3.5 jet periods and the fourth around 5. Each one of them has again greater amplitudes than the foregoing. It is also relevant to point out the almost invisible formation of a third wave packet at the

Reb = 600

case, which

originates almost at the same time that the downstream ow is breaking up into non-linearity, having thus a scarce and eeting linear development. Dierences between Reynolds numbers rest on the higher uctuation intensity growth, larger area affected, steeper gradients with the environment uid, and more abundant and tinier uctuation lobes, sign of a higher wavenumber; the greater Reynolds number gets. These more intense perturbations lead to an earlier transition to non-linear eects, characterised by the descomposition of lobes and formation of chaotic structures increasing perturbation in the whole domain. Another perceptible dierence, in terms of development between jets with dierent Reynolds number, is the variable creation of large lobes heading in several directions and aecting the entire domain. This is especially important with higher Reynolds number, but it can be observed even from

Reb = 600.

These consist in all cases, in the linear phase, in 6 lobes, one heading upstream, other downstream, and two towards each side.

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Between them, lines of lower uctuation are present.

73

They seem to be a sign


4


4.2


of emergent transition, since shortly after their generation non-linear eects happen. That is why with greater Reynolds number they appear earlier and with steeper gradients.

4.2.2 Maximum perturbation amplitude at each x-coordinate Figures 4.25 to 4.31 represent the maximum absolute values of the axial velocity perturbation at each

x-coordinate

of the jet.

On the independent axis the

x-coordinate

is presented, on the left-hand side

dependent axis the plain maximum absolute values are shown, and on the right-hand side one they are also displayed in logarithmic scale. Each gure corresponds to a dierent Reynolds number, ranging from the smallest to the greatest.

They include subgures for several time steps, expressed in jet periods,

ranging from 0.5 to 2.5 so as to comply with the previous section. However, an additional time step is presented, in which transition starts to occur. In this way a more accurate assessment of transition can be carried out.

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4


4.2


(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e)


(f )

Transition start, after 5.8 jet periods

Maximum absolute values of axial velocity perturbation with respect to x in natural and logarithmic scales at Reb = 300

Figure 4.25:

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4


4.2


(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods

(e)


(f )



Figure 4.26:

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4


(e)

4.2


(a)


(b)

After 1 jet period

(c)


(d)

After 2 jet periods


(f )



Figure 4.27:

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77


4


(a)


(c)


(e)

4.2


(b)

(d)

After 2 jet periods

After 1 jet period


(f )



Figure 4.28:

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78


4


(a)


(c)


(e)

4.2


(b)

(d)

After 2 jet periods

After 1 jet period


(f )



Figure 4.29:

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4


(a)

(c)

4.2



(b)

After 1 jet period


(d)


After 2 jet periods

(f )


(e)


Figure 4.30:

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80


4


(a)

4.2


(b)



(c)

After 1 jet period

(d)


(e)

After 2 jet periods

(f )



Figure 4.31:

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4


4.2


The formation and propagation of waves is checked by the previous gures, since they coincide with the eld results from section 4.2.1. The linear waves packets are distinctly observable since they correspond to higher amplitudes at a concrete tribution of maximum amplitudes.

x-span,

and the non-linear eects are characterised by a spread dis-

The coexistence of the two wave packets can be especially seen at

the 1.5 and 2 periods graphs, having the rst wave greater amplitudes at 1.5 periods and the other way

Reb = 1800 almost only non-linear eects are present. Dierences −7 at Re = 300 to 0.95 at numbers are also evident, ranging from 10 b

around at 2 periods. Nevertheless from in amplitudes between Reynolds

Reb = 2500

at 2.5 periods.

Transition. A deeper analysis of transition has been possible thanks to these maximum amplitude plots and animations, with the criterium that it is given when sinusoidal waves start to present non-linear irregularities after having developed. Therefore, we can present new transition time steps in Table 4.2, more accurate than the previous ones in Table 4.1, and the

x

coordinate in which they take place.

In order to have

comparable results, we specify the initial coordinate of the domain, that, remembering our domain setup from section 3.1.3, is dierent for each Reynolds number. However, domain's 85, corresponding to 85 initial widths

Reb

Transition

time

x

length is the same for all,

b.

Initial

x

coordi-

Transition

x coor-

Transition

step [jet periods]

nate (x0 )

dinate

300

5.8

1.4

50

57.18

600

2.9

2.79

50

55.54

800

2.2

3.72

32.5

33.86

1000

1.95

4.65

25

23.94

1200

1.75

5.58

80

87.55

1800

1.15

8.37

62.5

63.68

2500

0.8

11.63

55

51.02

span

in

x-

percentage

[%]

Table 4.2:

Transition in jet periods at each Reynolds number observed in maximum amplitude versus x plots

There exists some variation between these more accurate results and the aforementioned, though not steep. If we look at the transition location in terms of percentage, the comparable data, we perceive an earlier development of transition the greater the Reynolds number. Furthermore it is also obvious that the start of transition takes place due to the second wave packet at Reynolds numbers until 1000 (with the exception of

Reb = 1200.

Reb = 300,

in which it is due to the fourth), and due to the rst one at higher, from

It is also noticeable in the gures that the waves packets start to divide in several groups

after transition starts, having their own relative group maximums and being clearly separate from one another.

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4


4.2


Phase velocities, wavenumbers and frequencies. Following a graphic procedure in

R, Tecplot

phase velocities can also be obtained.

For this task it is

needed to distinctly declare the non-dimensional times corresponding to each jet period at each Reynolds number, previously calculated with

R. MATLAB

Table 4.3 summarises these values and in Figure 4.32

they are plotted.

Reb

Jet period time steps

Non-dimensional jet period

300

276000

235.2624

600

129000

192.4938

800

99000

177.2892

1000

84000

167.1348

1200

74000

159.4848

1800

60000

144.5760

2500

53000

134.2013

Table 4.3:

Figure 4.32:

Non-dimensional jet period at each Reynolds number

Non-dimensional jet period with respect to the Reynolds number

Due to the certain irregularity of the waves, the calculated magnitudes presented subsequently are not really accurate, since the wave behaviour is dierent throughout its domain crossing and even within the

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83


4


packet.

4.2


Therefore a standard position for every Reynolds number has been taken, in order to obtain

comparable results. The rst wave packet has been measured in the middle of the domain, and the second one at the rst quarter, owing to the quick development of it at high Reynolds number.

Wavelenghts

and periods have been obtained, and with them phase velocities, wavenumbers and frequencies, all in non-dimensional quantities, summarised in Table 4.4. First wave

Reb

Wavelength

Period

Phase

Wavenumber

Frequency

velocity 300

8

44,75

0,18

0,79

0,02

600

6

32,50

0,19

1,05

0,03

800

4,5

19,91

0,23

1,40

0,05

1000

4

11,69

0,34

1,57

0,09

1200

3,5

8,92

0,39

1,80

0,11

1800

2,75

6,04

0,46

2,28

0,17

2500

2,5

4,50

0,56

2,51

0,22

Reb

Wavelength

Period

Phase

Wavenumber

Frequency

300

9

57,54

0,16

0,70

0,02

600

7

32,50

0,22

0,90

0,03

800

4,5

19,91

0,23

1,40

0,05

1000

4,5

16,37

0,27

1,40

0,06

1200

3,5

8,92

0,39

1,80

0,11

1800

2,5

6,04

0,41

2,51

0,17

2500

2,5

5,62

0,44

2,51

0,18

Second wave

velocity

Table 4.4:

Wavelenghts, periods, phase velocities, wavenumbers and frequencies at each Reynolds number

In the light of the outcome, phase velocities are always higher with greater Reynolds number. Furthermore, in general the rst wave travels faster than the second one, the former having shorter periods, though also shorter wavelenghts or greater wavenumbers. As the Reynolds number increases and for both waves, phase velocities are more elevated owing to a shorter period, overcoming the lower wavelength or higher wavenumber. In consequence, the frequencies of the waves at higher Reynolds number are more elevated.

4.2.3 Contours of maximum amplitudes at each time step with respect to x-coordinate In Figures 4.33 to 4.39 the contours of logarithm of maximum absolute values of axial velocity uctuation at each time step, with respect to the spatial evolution of the jet, the

x-coordinate,

are plotted.

The

white continuous lines represent the movement of waves front, what provides us with information about their group velocity. The dashed lines represent the speed of sound in opposite direction to the waves, the so-called

-a

lines. These have dierent slopes at each Reynolds number, since the time scale is presented

in jet periods. As our Mach number is equal to 0.2, the speed of sound in non-dimensional quantities is

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4.2


equivalent to 5. The time it takes to cross the entire domain (85) is 17 at all the cases. However, owing to the shorter period, the speed of sound represents a higher slope the greater the

Reb .

Moreover, it is

important to remember that there are more periods represented in the two rst cases. Contours begin at -14 in logarithmic scale, and end in -1 until

Reb = 800

and in 0 for the rest, representing the whole

span of perturbation evolution. They are also represented in linear scale in order to see clearly the highly perturbed region and phases.

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4


4.2

(a)


In logarithmic scale

(b)

In linear scale

Contours of maximum absolute values of axial velocity perturbation at each x-coordinate wiht respect to time at Reb = 300

Figure 4.33:

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4


4.2

(a)



(b)

In linear scale


Figure 4.34:

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87


4


4.2

(a)



(b)

In linear scale


Figure 4.35:

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4


4.2

(a)



(b)

In linear scale


Figure 4.36:

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89


4


4.2

(a)



(b)

In linear scale


Figure 4.37:

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90


4


4.2

(a)



(b)

In linear scale


Figure 4.38:

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91


4


4.2

(a)



(b)

In linear scale


Figure 4.39:

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4.2


The preceding gures are quite revealing, in the sense that help to understand the generation of subsequent waves after the rst one.

Reb = 300,

3 at

Reb = 600,

The number of produced waves is distinctly noticeable, being 4 at

the last one being ephemeral; and 2 at the rest. We can now state with more

accuracy than with the perturbation eld, that the generation of the rst wave occurs immediately at the beginning of the calculation, of the second one around 1.2 jet periods, the third one around 2.75 and the fourth around 4.5. By following the trends of movement of wave fronts, it can be seen that the end of the preceding and the origin of the succeeding are interrelated by means of the regressive speed of sound line.

This is a

distinct sign that successive waves are generated at the commencement of the jet by virtue of the acoustic reection of the previous wave in the end boundary of the domain.

The pressure perturbation caused

by the interaction of the front of the preceding wave with the end limit bounces and travels back to the jet origin, since sound progagates in all directions. There, it triggers the new instability package, having greater amplitudes than the foregoing, but as we can see in the gures and in next paragraph, with the same group velocity. It is also discernible the quick increase in the order of magnitude of the maximum amplitudes as waves develop. Regarding non-linear eects, it is noticeable that at rst, at transition, they are given by slopes of greater velocity than the wave. These start regular but become irregular after some further development, producing a ow pattern with not only pretty fast regions but also pretty slow. The appearence of these acoustic reections is logical, since specic acoustic non-reecting boundary conditions have not been implemented. It is worth it to point out that these are a type of boundary condition still under development, being a dicult phenomenon to mitigate. Even more in a DNS, where the whole N-S equations are solved. Some examples of proposals of non-reecting conditions are the ones of Sandberg

et al,

[25], and Fajardo

et al

(2011), [26]; the former for DNS and the latter for CFD commercial

codes. In our case, acoustic reection is an inevitable process that occurs in our domain and determines considerably the behaviour of instability waves, triggering the generation of subsequent ones. Whether these instabilities are absolute or convective is also a feature easy to obtain from the previous

x -t

graphs.

It is evident, that these instability waves are convective, since with time they move away

from their initial position fast enough so that at a given longitudinal position the disturbance decays with time. This can also be checked in the rest of results gures, but since it is a relationship between spatial and temporal development, it is in these

x-t plots where it can be better analysed.

It is simply convective

because we can clearly observe that the group velocity of the waves is not zero, condition for this type of instabilities. If they were so they would stay at a xed

x

coordinate, being absolute instabilities.

Group velocities. From the previous graphs we can easily obtain the group velocity of each wave packet, since they correspond to the white lines plotted. It is appreciable that all the waves within each case share the same pattern of group movement, what in practical terms means an equal group velocity. The measurement of the white lines slopes has been carried out in second half of the domain, where waves are completely

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4.2


linearly developed and slopes are clearly identiable. At the formation phase, the location of the wave front is rather misty, what makes it a more dicult measurement at higher Reynolds number, where the second wave has a eeting linear existence. Those values are presented in Table 4.5 and plotted in Figure 4.40.

Table 4.5:

Figure 4.40:

Reb

Waves group velocity

300

0.21

600

0.32

800

0.40

1000

0.44

1200

0.48

1800

0.65

2500

0.75

Group velocities at each Reynolds number

Group velocities with respect to the Reynolds number

As logical, group velocities increase the greater the Reynolds number since the jet period is shorter, and the waves development is similar in terms of jet periods for all Reynolds regimes. It is also noticeable, that

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4.2


with this graphical calculation, a quite accurate linear relationship between group velocities and Reynolds number comes out.

4.2.4 Maximum amplitude at each y-coordinate for several x-locations Figures 4.41 to 4.47 show the logarithm of maximum absolute values of axial velocity perturbation with respect to the

y -coordinate

at ve

x

positions, ranging equidistantly from inlet to outlet. Amplitudes are

represented in the independent axis so as to facilitate the spatial visualisation with respect to the

y

axis.

Each gure corresponds to a dierent Reynolds number, ranging from the smallest to the greatest. They include subgures for two dierent time steps: 1 and 2 jet periods.

Figure 4.41:

(a)

After 1 jet period

(b)

After 2 jet periods

Logarithm of the maximum absolute values of axial velocity perturbation with respect to y at

Reb = 300

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4


Figure 4.42:

4.2

(a)

After 1 jet period

(b)

After 2 jet periods



Reb = 600

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4


Figure 4.43:

4.2

(a)

After 1 jet period

(b)

After 2 jet periods



Reb = 800

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4


Figure 4.44:

4.2

(a)

After 1 jet period

(b)

After 2 jet periods



Reb = 1000

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Figure 4.45:

4.2

(a)

After 1 jet period

(b)

After 2 jet periods



Reb = 1200

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4


Figure 4.46:

4.2

(a)

After 1 jet period

(b)

After 2 jet periods



Reb = 1800

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4


Figure 4.47:

4.2

(a)

After 1 jet period

(b)

After 2 jet periods



Reb = 2500

In these gures it is discernible that the main perturbed region is close to the centreline, in an area be-

−3b and 3b approximately for the early periods, and a bit wider for the late ones, reaching y -values ±5b for lower Reynolds numbers and almost ±10b for the higher non-linear ones. This is a logical result

tween of

since the main uctuation lobes are always seen all over this central region, main area aected by the jet. It is also common among the cases that amplitudes are higher towards the half or three quarters of the domain, where waves are completely developed and have grown in intensity. The temporal evolution is manifest too, all having steeper gradients in

y

at the second period.

Moreover, at the outlet there is generally a homogene distribution of amplitudes at the rst period, but at the second the central region is still more perturbed. At the inlet amplitudes are in all cases negligible in comparison to the rest of the domain, surely because of the constant pressure gradient boundary. With regards to the dierences, it is again patent the increase in orders of magnitude of the amplitudes and gradients with the environment uid with the Reynolds number, attaining values of order

Reb = 300

−1 at until 10

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Reb = 2500.

From

Reb = 1800 101

10−8

at

the orders of magnitude between the rst and


4


4.2

second period practically do not change, staying in ranges from

10−4

to


10−1

for the most active zones.

At lower Reynolds number there is always a dierence between periods of around 2 orders of magnitude. Also about these Reynolds numbers, it is perceptible a more regular and rounded distribution along the

y -axis.

From

Reb = 800

the second period distribution starts to be irregular, and from

Reb = 1200

the

rst one.

4.2.5 Field maximum amplitudes at each time step In Figure 4.48 the global maximum absolute values of velocity perturbation amplitudes at each time step in logarithmic scale are represented, for the whole Reynolds number span at the same time.

Figure 4.48: Global maximum absolute values of velocity perturbation amplitudes with respect to jet periods in logarithmic scale for the whole Reynolds number span

The previous graph shows us that there is a clear relationship between the Reynolds number and the perturbation evolution in a plane jet. At the beginning of all simulations there is a linear region in the gure, corresponding to an exponential growth of uctuation with time, with higher slope for higher Reynolds number but not very dissimilar among them. Main dierences appear from 0.25 periods, where the rates of growth start to diverge from one another, being lower for smaller Reynolds numbers. For Reynolds numbers until

Reb = 1200

another abrupt change of slope is observable, at dierent time

steps for each Reynolds number, but in a narrow region between and 1.75 at at

Reb = 300.

Reb = 1200 and 2.25 periods

This change corresponds to the dominance of the second wave packet over the rst, and

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4.2


therefore to the quicker growth of it. This means, that even though the second wave packet is always generated at approximately the same time step (in terms of jet periods) and location for all Reynolds numbers, its relationship with the rst wave varies, being more dominant the greater the Reynolds number. This is also an indication of the greater exponential amplitude growth of the second wave. After this slope variation, development ends up with a non-linear state uctuating around an order of magnitude of

10−1 , what corresponds to non-linear saturation.

For Reynolds numbers up from

Reb = 1800,

the second slope change almost does not appear, having an earlier transition to non-linearity. Therefore we can state that the second wave packet develops almost entirely non-linearly here.

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CONCLUSIONS

5 Conclusions In this section the key aspects of this Bachelor's thesis are summarised and related with one another, drawing conclusions. To sum up, Direct Numerical Simulation (DNS) has been employed to simulate a two-dimensional free plane air jet at dierent Reynolds numbers. The jet has been modelled with

R MATLAB

by programming

a Graphical User Interface (GUI) (presented in section 3.1.6) which basically creates a domain and a mesh and calculates the velocity, pressure, density and temperature elds (a complete baseow) according to the input data. These are computed by means of the Schlichting's self-similar plane jet equations (1933), from his Boundary Layer Theory, derived in section 3.1.1. This theory fundamentally assumes a constant moment ux at any cross section of the jet, since it propagates at constant pressure and there are no restraining walls. In addition, the self-similarity variable dened by him relates the dimensions through a non-linear expression. The type of computational method was chosen to be DNS since the validity of this self-similar model wanted to be evaluated as an initial guess to the Navier-Stokes equations. Moreover, this method gives the most accurate result thereby increasing the value of the results, and therefore having more scientic worthiness of the observed eects of the Reynolds number on this type of jets. The most important of the input data mentioned is the Reynolds number (dened in section 3.1.2), var-

Reb = 300 to Reb = 2500, passing through Reb = 600, Reb = 800, Reb = 1000, Reb = 1800, so as to check its inuence in the jet development and characteristics. Our number includes the maximum axial velocity at jet emergence umax , the characteristic length

ied within a range from

Reb = 1200 Reynolds

b,

and

this being the jet width in which axial velocity decays to a 1% of its maximum centreline value; and

density and dynamic viscosity corresponding to quiescent air at International Standard Metric Conditions. The

b

parameter is the only one varied with the Reynolds number, since the maximum velocity or

in other words the Mach number is kept constant in order to maintain the same compressible behaviour in our solution. In practical terms, this length can also be considered as the jet diameter at its origin, since the domain starts at the

x-coordinate

in which the condition for

b

is given, having

centreline. It is important to point out that this characteristic length is very small, of order

umax at the 10− 4 meters

in dimensional coordinates (tenths of milimeters), since the Reynolds numbers selected are quite low in order to visualise well the instability formation at laminar regime. fullled and a variation of

Reb

This way self-similar equations are

can be carried out maintaining a constant Mach number, equal to 0.2 all

over this thesis, ensuring a laminar start of the jet. The eects of this Reynolds number variation are manifest in the results and produce the expected trends of earlier transition and more intense uctuations, what validates our approach. The rest of input data for the GUI are the following, mainly dening a common frame for every so as to obtain contrastable results. widths

b

in

x

and 60 in

y,

Reb

case

First of all, the domain dimensions have been adjusted to 85 jet

with an irrelevant

z -length

due to our two-dimensional approach, as explained

in section 3.1.3. These dimensions were chosen in the interest of contemplating all the phases of jet development in a comprehensive way, not only entrainment in transversal direction but also velocity decay and spatial evolution in the longitudinal one. In the end it has been corroborated that this size is large enough to observe instability generation and development and transition processes, by accomplishing the

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CONCLUSIONS

previous visualisation requirements. Concerning the meshing input parameters, presented in section 3.1.4, mainly two need to be recalled. These are the number of subdomains in which the whole domain is divided to reduce calculation times thanks to parallelisation, and the number of nodes of the grid in

x

and

y

axes. A suitable ansatz was

veried to be a 24-subdomain eld (6 along axial direction and 4 along normal) with 200 gridpoints in direction and 275 in

1.32 × 106

y

x

per subdomain, achieving 1200x1100 in the overall domain, which means a total of

points. With more points, results came out to be identical, sign of having reached an appropri-

ate number of nodes. Moreover, the meshing strategy in terms of size and distribution of this amount of nodes, non-modiable by the GUI user, was seen to be adequate by implementing three dierent element sizes.

They have higher concentration in the jet core (one third of the

y

span), and less at the outer

regions, were gradients are less important and not so much precision is required. This way considerable computing time was saved. On the basis of this mesh, the

R MATLAB

programme calculates the aforementioned elds in non-

dimensional quantities, required by the DNS code used to simulate the jet. The velocity eld is normalised by

umax ,

the spatial variables by the characteristic length

b

and the rest of elds by their freestream ref-

erence values, as explained in more detail in section 3.1.5. With regards to the boundary conditions chosen, enumerated in section 3.2.1, they have been observed to function well and express adequately the kind of ow desired. At the inlet, the condition is dened by a zero pressure gradient in the

x

axis with extrapolation of pressure, since it is not a pressure driven

ow. At the outlet a zero-gradient one is implemented, dening all the eld variables as equal as in the previous

x coordinate.

The lateral boundaries, presented a satisfactory behaviour when simply considered

as freestream. This implies zero gradients in

y

axis all over the limits.

Another determinant tool employed while calculating was the damping zone or

sponge (analysed in section

3.2.2), implemented at the last 5% of points of the domain, with a gain of 0.5, since not really strong attenuating eects were needed. The necessity for this damping zone was noticed while analysing results in terms of axial velocity perturbation, where values start being some order of magnitude lower than the base-state velocity. Huge computational rebound eects of the end boundary were present, aecting the entire domain in a considerable manner. With the damping zone results came out to be more accurate, stable and compliant with a real ow. While testing with this instrument, an interesting highly disturbing computational interaction was discovered. This consisted in the interaction between the damping zone and a random perturbation included in the baseow velocities so as to promote the instability formation in the jet.

When this perturbation was taken out, results started to be better in terms of linearity of

the instability wave packets. The causes for this phenomenon are left to the reader for further research, though probably they may be related to the ineciency of the

sponge

against a highly perturbed ow.

With respect to the simulation performance parameters, introduced in section 3.2.3, it is worth to say that it was always checked that the Courant-Friedrichs-Lewy condition (CFL) or security factor was close to the unity, a bit lower though. This task could be performed with the help of short test simulations to analyse its behaviour, and of the

R MATLAB

GUI, which contains a section in which the simulation

performance parameters, such as fundamental frequency, time steps and period of time steps can be easily

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CONCLUSIONS

dened. These data are obviously in conjunction with the amount of time that the jet is evolving. In order to make easier comparisons among the dierent Reynolds number cases, the concept of jet period was dened. A jet period expresses the non-dimensional time needed for the passage of a uid particle from the centreline since it is exhausted in the nozzle until it leaves the domain, at initial conditions. The time steps of each simulation were chosen according to this magnitude, being 7 periods for the case, 4.2 for the

Reb = 600,

Reb = 300

and 2.8 for the rest, all of them consisting in development phases after

transition. The minimum of 2.8 was chosen because it corresponded to the time steps that the computing system could calculate in 24 hours at the

Reb = 600

case, which was the one used to obtain the optimum

computational setup. Transition was not given in this interval at

Reb = 600

but virtually, so for the rest

of cases it was ensured. An additional option of the IAG's

NS3D

DNS code consisted in the activation of the disturbance calcu-

lation (claried in section 2.4.2) instead of the standard one. It was decided to adopt it because of the high orders of magnitude of uctuation distributions and relatively early transition, not being suitable for the subsequent instability postprocessing. Moreover, disturbance calculation is more coherent with our objectives related to the initial self-similar solution, inasmuch as this ansatz adopts the given baseow as a steady-state solution of the Navier-Stokes equations. Therefore we are also checking the deviations of the developed ow from the proposed solution. About the computational resources exploited, 24 MPI (Message Passing Interface) processor boards, one for each subdomain, have been available for parallelisation in order to enhance the simulation speed, as described in section 3.2.4. Each of the boards could be used with its maximum calculation power, that is 32 OpenMP (Multi-Processing) threads. With this calculation resources, simulations could be performed in around 17 hours per jet period at the lowest Reynolds number, and around 3,5 at the highest, since the number of time steps necessary for low time steps per period for

Reb = 300

Reb

cases is considerably higher. These range from 276000

to 53000 for

Reb = 2500.

As more than one period was needed to

study transition, computing times reached 5 days at the lowest Reynolds case and 9,5 hours at the greatest. Three-dimensional tests were also tried, as commmented in section 3.2.5, with no positive results in terms of third direction development without including high random perturbation, and not being really coherent with Schlichting's self-similar model. Regarding the DNS code characteristics, described in detail in section 2, it is relevant to recapitulate the following points. First of all, the unsteady compressible Navier-Stokes equations are formulated in the programme three-dimensionally and in a conservative and non-dimensional way. Secondly, it presumes the uid to be a non-reacting ideal gas following the equation of state.

About the numerical method,

the simplest spatial discretisation approach has been selected, that is Compact Finite Dierences of 6

th

order, owing to the fact that we are dealing with a not especially large two-dimensional mesh simulated within a relatively short time for DNS. In addition the time integration is carried out by means of a 4

th

order Runge-Kutta Method. Up to this point we have recapitulated every parameter selection of the computational setup for an optimum visualisation of jet development, which we were able to reach within the available time. Moving on to the analysis of results, we will comment on the axial velocity contours at rst, illustrated in section

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CONCLUSIONS

4.1. There we could see a dependence on the Reynolds number in terms of intensity of instabilities (amplitudes) and transition, not really precisely though. At the beginning of every simulation an apparent stationary phase with no deviation from the self-similar baseow solution seemed to take place, the longer the lower the Reynolds number. Nevertheless, this is only appearance, because, as we saw in the velocity uctuation section, there are mainly two linear sinusoidal instability wave packets that can be partly identied in the whole axial velocity eld, but perceived in more detail in the perturbation eld.

The

reason for that is that the amplitudes of these waves at start is some orders of magnitude lower than the jet core velocity. However, it is manifest that the direct consequence of greater dominance of the convective terms against the viscous ones is the generation of intense waves closer to the jet start and so at earlier time steps. Furthermore, transition occurs earlier with a Reynolds number increase, characterised by the corruption of the linear sinusoidal waves, and formation of non-linearity. Hence, some parts of the domain end up with even negative values of axial velocity, the lower the Reynolds number. By looking at these elds in an exponential scale, it is also patent that the whole domain is aected by instabilities at the jet, provoking its achievement of orders of magnitude of 0.1 or 1 in the whole eld. In addition, it discernible that jet development and entrainment of ambient uid are connected, with the outcome of several phases of instability generation at the nozzle exit and amplication all over the jet core, reaching transition into non-linearity. With regards to the more interesting axial velocity perturbation elds, exhibited in section 4.2.1, it is appreciable that the generation of these wave packets of perturbation occurs at approximately the same jet period steps and positions at all the Reynolds numbers. A conclusion that can be drawn from this fact, is that the generation of these waves hinges upon the relative development of each jet, in terms of the time required for a centreline particle to cross the domain. A common phenomenon between the cases from

Reb = 600

to

Reb = 2500,

is the explicit distinction of

two wave packets before transition. However, the wave triggering transition is not the same, for Reynolds numbers until 1000 the second one does, and from

Reb = 1200

it is the rst one.

But in general the

second wave ends up with greater amplitudes and gradients. The

Reb = 300 case is fairly special, since more waves spring before its transition.

The amount of observed

waves is four, and the last two appear at the end of the prior wave. Each one of them presents again greater amplitudes than the preceding. In general for all cases, the formation of waves has been checked with the

x

to

t

plots to be at the begin-

ning of the calculation for the rst one, around 1.2 jet periods for the second, around 2.75 for the third and around 4.5 for the fourth, the last two being only characteristic of the low Reynolds number regimes. Main dierences between the studied cases consist in the higher uctuation intensity increase, larger surrounding area disturbed, more abrupt gradients with the ambient uid, and more abundant and smaller uctuation lobes, indication of a higher wavenumber; the greater Reynolds number. These more intense perturbations are responsible of the aforementioned earlier transition.

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CONCLUSIONS

By performing a deeper analysis of these elds, in terms of maximum amplitudes with respect to several variables, interesting trends could be seen. At rst with respect to the longitudinal axis, in section 4.2.2, dierences in amplitudes between Reynolds numbers are evident, ranging from at

Reb = 2500

10−7

at

Reb = 300 to 0.95

at 2.5 periods.

Transition was studied in depth, in terms of time and location, conrming the perceived earlier development the greater the dominance of the convective terms in opposition to the viscous ones. Furthermore it is also corroborated that transition occurs due to the second wave packet at Reynolds numbers until 1000 (with the exception of from

Reb = 1200.

Reb = 300,

in which it is due to the fourth), and due to the rst one at higher,

Waves division in several groups after transition starts is also evident.

If we treat the important matter of the causes that provoke the generation of instability waves, we can denitely state that the main stimulation for the formation of the rst wave is the adaptation of the approximated model of the Navier-Stokes equations solution that we have introduced to the entire actual equations. This adaptation results in creating a perturbation that travels with the jet. The reason for the successive waves has been identied to be acoustic reection of wave fronts in the end boundary condition, thanks to the

x

versus

t

plots of the contours of maximum amplitudes, illustrated

in section 4.2.3. The indication for this phenomenon is the connection of the front of the preceding wave with the one of the following via the regressive speed of sound characteristic. The presence of this process is inevitable in our circumstances, since special acoustic non-reecting boundary conditions should have been developed for this purpose. It is certain though, that these reections shape the evolution of the jet ow, allowing the generation of subsequent waves and succeeding transition into non-linearity. While measuring group velocities, it could be seen that they increase the greater the Reynolds number since the jet period is shorter, and the development of the waves is equivalent for all Reynolds cases. The fact that these velocities are not zero, is a sign that the instabilities we are dealing with are of convective kind, that is to say, that they travel with the ow, with a disturbance decay behind them. Phase velocities are also always higher with greater Reynolds number. Frequencies of the waves at higher Reynolds number are likewise more elevated. Furthermore, the rst wave travels faster than the second one, the former having shorter periods, though also shorter wavelenghts or greater wavenumbers. Nevertheless, these magnitudes are not constant throughout the waves evolution. Therefore these waves cannot be considered as periodic. This characteristic did not allow to perform a Fast Fourier Transform postprocessing. About the results in term of

y

axis dependence, illustrated in section 4.2.4, it is notable that the main

disturbed region is near to the centreline, in an area between periods, and it widens with time, reaching

y -values

of

±5b

−3b

and

3b

approximately for the early

for lower Reynolds number and almost

±10b

for the higher non-linear ones. It is also similar among the cases that amplitudes are higher towards the half or three quarters of the domain, where waves are completely developed and have enlarged in intensity. Again, increase in orders of magnitude of the amplitudes and gradients with the Reynolds number is patent. A particularly interesting graph, the one dealing with eld maximum amplitudes at each time step pre-

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CONCLUSIONS

sented in section 4.2.5, demonstrates a clear relationship between the Reynolds number and the perturbation evolution in a plane jet. At the beginning of all simulations there is a linear region in the gure, corresponding to an exponential increase of the perturbation with time. From 0.25 periods, the rates of increase begin to diverge from one another, being lower for smaller Reynolds number. A steep change of slope, matching with the dominance of the second wave packet over the rst, is given at Reynolds numbers

Reb = 1200, at dierent time steps, but in a narrow region between and 1.75 at Reb = 1200 and 2.25 periods at Reb = 300. This implies that despite the similar generation of waves in terms of jet periods,

until

their relationship with one another varies, the second being more dominant the greater the importance of the convective terms counter to the viscous ones. This is also a sign of the higher exponential amplitude growth of the second wave. At the nal part of development, jets end up with a non-linear state uctuating around an order of magnitude of from

Reb = 1800,

10−1 ,

considered as non-linear saturation. For Reynolds numbers up

the second slope change is negligible, having an earlier transition to non-linear regime,

and eeting linear development of the second wave. All in all, we can state that the computational setup selected has been adequate for an instability and transition analysis, leading to coherent results in terms of eects of the Reynolds number. The validity of the Schlichting's model (1933) as a DNS initial guess at the low Reynolds number span calculated, at constant Mach number of 0.2, has been corroborated. This verication is mainly based on the existence of a considerably long phase of linear development of the jet velocity eld, having though an inevitable transition into non-linearity due to the initial corrective perturbation and its subsequent acoustically reected waves. Further research of this computational frame is left to the reader.

Interesting proposals would be, for

instance, investigating the eect of a dierent Mach number and so dierent maximum velocity, instead of the jet width, or performing 3D simulations with long computing times, in order to visualise some spanwise velocity perturbation. Moreover the eects of the inclusion of random perturbation at the baseow and its interaction with damping zones are also intriguing issues not puzzled out within this thesis.

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CONCLUSIONS

Appendix 1: Simulation les links In this appendix we include the at 10 fps in

mp4

Dropbox

links to the

R MATLAB

GUI scripts and to video animations

format corresponding to all the simulations, in terms of axial velocity elds in linear and

exponential scale, and axial velocity perturbation elds.

Complete .zip packet •

R GUI scripts and video animations): https://dl.dropboxusercontent. .zip packet (MATLAB com/u/110113914/Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca.zip

Complete

R MATLAB

GUI scripts

•

.g le: https://dl.dropboxusercontent.com/u/110113914/Bachelor%27s%20Thesis%20Sergio% 20P%C3%A9rez%20Roca/MATLAB%20GUI/freejet.fig

•

.m le: https://dl.dropboxusercontent.com/u/110113914/Bachelor%27s%20Thesis%20Sergio% 20P%C3%A9rez%20Roca/MATLAB%20GUI/freejet.m

•

Additional

.txt

le for computing speed enhancement, containing intermediate calculation values

300 5 Reb = 10000: https://dl.dropboxusercontent.com/u/110113914/Bachelor%27s% 20Thesis%20Sergio%20P%C3%A9rez%20Roca/MATLAB%20GUI/Saved_X_0_for_Reynolds_300-10000. txt

for

Reb = 300 •

Axial velocity eld in linear scale: https://dl.dropboxusercontent.com/u/110113914/Bachelor% 27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re300/Re300_u.mp4

•

Axial velocity eld in exponential scale:

•

Axial velocity perturbation eld in exponential scale:

https://dl.dropboxusercontent.com/u/110113914/ Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re300/Re300_u_exponential.mp4 https://dl.dropboxusercontent.com/u/ 110113914/Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re300/Re300_u_perturbation. mp4

Reb = 600 •


•


https://dl.dropboxusercontent.com/u/110113914/ Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re600/Re600_u_exponential.mp4

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CONCLUSIONS

•

https://dl.dropboxusercontent.com/u/ 110113914/Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re600/Re600_u_perturbation. mp4


Reb = 800 •


•


•



Reb = 1000 •


•


•



Reb = 1200 https://dl.dropboxusercontent.com/u/110113914/Bachelor% 27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re1200/Re1200_u.mp4

•

Axial velocity eld in linear scale:

•


•



Reb = 1800 •

https://dl.dropboxusercontent.com/u/110113914/Bachelor% 27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re1800/Re1800_u.mp4 Axial velocity eld in linear scale:

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CONCLUSIONS

https://dl.dropboxusercontent.com/u/110113914/ Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re1800/Re1800_u_exponential.mp4

•


•


https://dl.dropboxusercontent.com/u/ 110113914/Bachelor%27s%20Thesis%20Sergio%20P%C3%A9rez%20Roca/Re1800/Re1800_u_perturbation. mp4

Reb = 2500 •


•


•



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REFERENCES

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Experimental Investigations of The Inuence of Reynolds Number and Boundary Conditions on a Plane Air Jet PhD Thesis, The University of Adelaide, 2005 http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDAQFjAB&url= http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F235948738_Experimental_ Investigations_of_The_Influence_of_Reynolds_Number_and_Boundary_Conditions_on_ a_Plane_Air_Jet%2Ffile%2F72e7e514a37768ddb6.pdf&ei=WlDiUorME8PZtQa2poGwBA&usg= AFQjCNHBL3HFVb_IT2cH41IFBM6HnGLWEQ&bvm=bv.59930103,d.Yms&cad=rja [2] Schlichting, H.

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Recent Advances in Turbulence Hermisphere, New York, chapter The Self-Preservation of Turbulent Flows and Its Relation to initial conditions, pp. 39-73., 1989 http://onlinelibrary.wiley.com/doi/10.1002/aic.690360125/pdf [12] Comte, P., Lesieur, M., Laroche, H. and Normand, X.

Numerical simulations of turbulent plane shear layers Turbulent Shear Flows 6 (ed. J.-C. André, J. Cousteix, F. Durst, B. E. Launder, F. W. Schmidt and J. H. Whitelaw), pp. 361-380. Springer. 1989 http://link.springer.com/chapter/10.1007%2F978-3-642-73948-4_29 [13] Stanley, S. and Sarkar, S.

Simulations of spatially developing plane jets AIAA Paper 97-1922, 1997 http://arc.aiaa.org/doi/pdf/10.2514/6.1997-1922

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A numerical investigation on the eect of the inow conditions on the self-similar region of a round jet Phys. Fluids 10(4), 899-909., 1998 http://scitation.aip.org/docserver/fulltext/aip/journal/pof2/10/4/1.869626.pdf? expires=1392053678&id=id&accname=373871&checksum=55FB7E7076A93B34C864DF8735D8FD50 [16] Le Ribault, C., Sarkar, S. and Stanley, S. A.

Large eddy simulation of a plane jet Phys. Fluids 11, 3069-3083, 1999

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A study of the ow-eld evolution and mixing in a planar turbulent jet using direct numerical simulations J. Fluid Mech. 450, 377-407, 2001 http://aa.dlut.edu.cn/doc/Homepages/GUAN_Hui/Jet-paper/p377.pdf [18] Klein, M., Sadiki, A. and Janicka, J.

Investigation of the inuence of the reynolds number on a plane jet using direct numerical simulation Int. J. Heat and Fluid Flow 24, 785-794, 2003 http://ac.els-cdn.com/S0142727X03000894/1-s2.0-S0142727X03000894-main. pdf?_tid=fe31d5e6-9278-11e3-b9f6-00000aab0f01&acdnat=1392053591_ b0f84c27d4b21e76b2d3c8127d5d75df [19] CFD Online

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Direct Numerical Simulation of Noise-Generation Mechanisms in the Mixing Layer of a Jet PhD Thesis, Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, 2009

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