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JOINT INSTITUTE FOR AERONAUTICS AND ACOUSTICS

NI\S/\ National Aeronautics and Space Administration

Ames Research Center

NASA-CR-166185 19810012485

JIAA TR-26

KINEMATICS AND DYNAMICS OF VORTEX RINGS IN A TUBE

JAMES G. BRASSEUR

STANFORD UNIVERSITY Department of Aeronautics and Astronautics

Stanford, Ca lifornia 94305

~ ... , , -

Stanford University

... ___ al!!llJ!!!!l' ~ 1, 1 - ,., \ ~

AUGUST 1979

https://ntrs.nasa.gov/search.jsp?R=19810012485 2020-07-01T21:00:29+00:00Z

JIAA TR - 26

KINEMATICS AND DYNAMICS OF

VORTEX RINGS IN A TUBE

JAMES G. BRASSEUR

AUGUST 1979

The work here presented has been supported by the Nat10nal Science Foundation under Grant No. ENG 74-226 1 5, by the Department of Aeronautics and Astronautics, and by the Joint Institute for Aeronautics and Acoust1CS.

(The natural) laws are not forces external to things,

but represent_the harmony of movement immanent in them.

-from the I Ching.

ii

)\bstract

Although vortex rings have been studied mathematically and

experimentally for over a century, the transition in characteristics

from viscous to turbulent vortex rings, and the effects of boundaries

on their evolution is not well understood. In this study kinematic

theory and flow visualization experiments are combined to examine the

dynamic processes which control the evolution of vortex rings from

very low to very high Reynolds numbers, and to assess the effects of

the wall as a vortex ring travels up a tube.

THEORY. We model the vortex ring as a circular vortex filament

axisymmetrically placed ~n an infin~tely long tube. Using the

B~ot-Savart law, a k~nemat~c expression wh~ch l~nearly relates a given

distr~but~on of vort~city to ~ts veloc~ty field, the closed form

solutions to the non-steady potential and streamline fields are

obta~ned. Transferring to a frame mov~ng with the ring, the outermost

closed streamline outl~nes the sphero~dal volume of fluid carried

along w~th the vortex core. We find that the effect of the tube on

the vortex r~ng is to increase its volume and decrease its speed as

compared with an ~dent~cal ring in an unbounded flow. By

superpos~t~on of vortex f~laments, we conclude that the size and shape

of typ~cal vortex rings are accurately predicted by concentrating

their strength along a circle, and by neglecting vorticity in a wake.

EXPERIMENT. Vortex r~ngs are produced by eject~ng a single

pulse of water through or~fices mounted in a plexiglass tube of

d~ameter 11.88 cm. Seven orifices with or~fice-to-tube diameters

between 0.10 and 0.64 are used. The piston speed is adjusted so as to

produce tube Reynolds numbers of roughly 5000, 4000, and 3000,

resulting in the production of vortex r~ngs w~th Reynolds numbers

between 690 and 50 1 00. To visualize the flow, dye and hydrogen

iii

bubbles are used, high speed 16mm mOV1es record1ng all comb1nat1ons of

tube Reynolds number and or1f1ce size. From the mOV1es detailed

measurements are made of the d1stance, size, and shape as a funct10n

of t1me for 26 rings at d1fferent Reynolds numbers. A careful

examinat10n of the data reveals:

1. The t1me dependent form of the r1ng veloc1ty can be 1dentif1ed

w1th1n reg1mes for all vortex r1ngs, two regimes being separated

by a rapid change 1n velocity.

2. From the form of t1me dependence in velocity, three classes of

vortex r1ngs can be 1dentif1ed and grouped by Reynolds number.

Qualitat1ve observations are combined with the measurements to study

the character of vortex r1ngs 1n each of these three groups and the

per10ds of rap1d change between regimes.

COMBINATION OF THEORY WITH EXPERIMENT. The k1nematic

relat10nships among the size, shape, speed, and strength of vortex

rings 1n a tube are computed from the theory. Prev10us methods for

computat1on of the total c1rculation requ1red a measurement of the

veloc1ty field within the vortex ring. Together with these k1nematic

relationsh1ps, however, relatively simple flow v1sualizat1on

measurements can be used to calculate the total clrculation of a

vortex ring at a given time. Using thlS method we have computed and

plotted the strength as a function of tlme for our exper1mentally

produced vortex rlngs. Reynolds number relatl0nshlps are establlshed

and quant1tative differences among the three Reynolds number groups

are discussed. The accuracy of the method, assessed by comparing w1th

the measurements of Sullivan et al., 1S felt to be excellent.

iv

Ack11owledgeme11ts

It is very d~fficult in the course of one's work to express

one's true feelings of grat~tude and appreciation to all those who

have helped along the way. Instead, we are left at the end with the

page called "acknowledgements." I only hope that those who have been

so good to me over the past five years realize that my thanks go much

deeper than I can possibly express here.

I feel very fortunate to have had as my adviser Dr. I-Dee Chang.

Both a fine sc~entist and a wonderful person, he has taught me, among

many other th~ngs, that the po~nt of research is to learn.

For the help and support given me by Dr. K. Karamcheti and the

Joint Inst~tute for Aeronautics and Acoustics, ~ am truly grateful.

The Institute's act~v~t~es have been a constant source of stimulation

and knowledge.

I thank those who have g~ven me their technical expert~se and

help. Dr. Sotiris Koutsoyann~s played a major role in the design of

the exper~mental apparatus and in the execut~on of the experiment; the

apparatus was built by Vad~m Matte and Jerry DeWerk. Bill Janeway

designed and bu~lt the t~mer used to calibrate our movies as well as

other c~rcuitry, and Jack MacGowen helped with the design of the

hydrogen bubble v~sualization.

Much of this work would not have been possible without the

generous help and computer support given me by Dr. John Wilcox and his

solar physics group ~n the Institute for Plasma Research. A warm

thanks to Dr. Philip Scherrer for the many hours he has spent teaching

me about computers, and to Eileen Gandolfin~ for her help, and her

sm~le.

v

So many others have been so helpful, and so kind. Theresa

Storm, Susan Triefenbach, M1nn1e Pas1n, Ella ZV1gna, Joyce Parker,

Mar1bel Quesada, Carolyn Edwards, Cherry Bob1er, Dick Remmel, Susan

Richardson, J1ll Izzarell1, ... thank you all!

The first three years of th1s research were supported by the

Nat10nal SC1ence Foundat10n under Grant No. ENG 74-22615. Add1tional

support has come from the Department of Aeronautics and Astronaut1cs

in the form of a Departmental Research Ass1stantship, and from the

Joint Inst1tute for Aeronautics and AcouSt1CS.

vi

1

2

3

Abstract Acknowledgements List of Illustrations List of Symbols

INTRODUCTION

1-1 Historical Outline 1-2 Objectives and Organization

BACKGROUND

2-1 Kinematics of Vorticity 2-2 Dynamics of Vorticity 2-3 Vortex Rings in an Ideal Fluid

2-4

2-5

2-3.1 2-3.2

The Classical Vortex Ring Thick Core Vortex Rings

Vortex Rings in a Real Fluid

2-4.1 2-4.2

Vortex Ring Formation Vortex Ring Evolution

Concepts Behind the Present Study

THEORY

3-1 Solution for a Circular Vortex Filament in a Tube

3-1.1 3-1.2 3-1.3

Harmonic Expansion for the Unbounded Ring Monopole Solution: The Green Function The Potential and Stream Functions in Non-Steady Coordinates

C011te11ts

iii v

ix xii

1

1 4

6

6 10 12

12 18

19

19 20

25

32

32

32 36

37

3-2 The Potential and Streamline Fields in the Fixed Frame 39

3-2.1

3-2.2 3-2.3

Inner and Outer Expansions for the Unbounded Ring Expansions for the Induced Field The Potential and Streamline Fields

39 40 42

3-3 The Velocity of the Vortex Ring in a Tube 48

3-4 The Potential and Streamline Fields in the Moving Frame 54

3-4.1 The Kinematic Relationships Among Vortex Ring Parameters

3-4.2 The Effect of a Finite Core on Ring Shape

vii

54 66

4

5

6

EXPERIMENT

4-1 Apparatus and Experiment 4-2 Qualitative Observations 4-3 Quantitative Results

4-3.1 4-3.2 4-3.3

Data Acquisition Vortex Ring Velocity Vortex Ring Radius and Thickness

4-4 Discussion

4-4.1 4-4.2 4-4.3

Reynolds Number Dependence Periods of Rapid Change: Instabilities Additional Observations

COMBINATION OF THEORY WITH EXPERIMENT

5-1 Total Circulation as a Function of Time 5-2 Accuracy of the Computation Method

5-2.1 5-2.2

The Effect of Wake Vorticity on Ring Shape Comparisons with Measured and Estimated Values

5-3 Vortex Rings and the Reynolds Number 5-4 Arterial Stenoses and the Wall Pressure

SUMMARY

APPENDICES

A B

Vortex Ring Trajectory, Size, and Shape Plots Vortex Ring Strength and Velocity Plots

BIBLIOGRAPHY

viii

70

70 76 81

81 85 90

92

92 95 97

102

102 109

109 110

115 121

130

135 187

206

2-1

2-2

2-3

2-4

2-5

2-6

2-7

2-8

3-1

3-2

3-3

3-4

3-5

3-6

3-7

3-8

3-9

3-10

3-11

3-12

3-13

List of IIlustratiol1s

FIGURES

Definitions for a vortical flowfield

Definitions for a vortex tube or filament

The cylindrical coordinate system

A vortex filament-loop

The classical vortex ring in steady and non-steady coordinates

The velocity and vorticity distributions in a real vortex ring

Laminar and turbulent vortex rings

Ring Reynolds numbers previously covered in experimental studies of vortex rings

Definitions for a circular vortex filament axisymmetrica11y placed in a tube

Representing a circular vortex filament as a disk of dipoles of uniform strength

The streamlines and constant potential lines in the fixed frame for an unbounded vortex ring with £ = 0.40

The streamlines and constant potential lines in the fixed frame for a bounded vortex ring with £ = 0.40

The wall-induced streamline and constant potential lines in the fixed frame for a bounded vortex ring with £ = 0.40

"'-

The Domb-Sykes plot for U. ~

The wall-induced velocity of a vortex ring in a tube

The analog of a vortex pair in a channel

The streamlines and constant potential lines in the moving frame for a bounded and unbounded vortex ring with £ = 0.40

The streamlines and constant potential lines in the moving frame for a bounded and unbounded vortex ring with £ = 0.70

The variation of T/R with Uo f~r different values of £

The variation of R'/R with Uo for different values of £

Constant potential lines for a vortex ring with a finite core compared with a circular vortex filament

ix'

page

8

8

14

14

17

21

23

26

33

33

45

46

47

51

52

53

58

61

64

65

68

4-1

4-2

4-3

4-4

4-5

4-6

4-7

4-8

4-9

4-10

4-11

5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

5-11

FIGURES (continued)

The basic experimental apparatus

The main components of the experiment

Piston trajectory characteristics

Photograph of a vortex ring at the highest Reynolds numbers

Refraction corrections: apparent vs. real radius

The velocity distribution for Run 9

The least squares fit slopes and residuals vs. point number for Run 9

Repeatability: comparison in trajectory between Runs 9 and 10

Correlation of sudden velocity changes with instabilities

The rocking oscillation observed in Run 5

Rocking oscillation: velocity plot for Run 5

The variation of T/R' with R'/po for different values of Uo

Comparison of mean velocity and strength with point by point calculations for Run 1

The streamlines for a vortex ring with a wake compared with a wakeless vortex ring

Comparison of vortex ring strengths computed using the kinematic theory with estimates and one measurement

Relationship between ReR

and Rer for newly formed vortex rings

Relationships between ReR or Rer and Re. for newly formed vortex rings J

The drag coefficient of vortex rings plotted against Reynolds number

Speed and Reynolds number of blood ejected from a human heart into the aorta

The non-dimensional wall velocity as a function of z/Po for different values of €

The pressure at the tube wall 1.5 diameters from the orifice for Run 1

The pressure at the tube wall 1.5 diameters from the orifice for Run 3

x

page

71

73

75

78

84

86

88

91

96

98

99

104

107

111

114

117

118

122

124

126

128

129

FIGURES IN THE APPENDICES

A-I through A-26

Vortex ring trajectory, size, and shape plots for Runs 1 - 26

B-1 through B-17

3-1

3-2

Vortex ring strength and velocity plots

TABLES

Numerically computed values for M n

Coefficients for the wall-induced velocity of a vortex ring in a tube

4-1 General identification for experimental

A-I

A-2

B-1

vortex ring runs

TABLES IN THE APPENDICES

Summary of experimental values for each run

Summary of regime slopes for each run

Initial vortex ring parameters for each run

xi

page

135

187

43

50

82

136

137

188

a

d

D

10, II

Jo, Jl

Ko, Kl

n -+ P, P

P -+ r

R

R'

t

T -+ u

u, u z

u p

u. J

U

'" U

V

V • -+eJ x

x (t) p

X P

x r

z

List of Symbols

LATIN SYMBOLS

radius of the core cross-sectional area (Fig. 2-3)

diameter of the orifice

inner diameter of the tube

modified Bessel functions of the first kind

Bessel functions of the first kind

modified Bessel functions of the second kind

normal to a surface

total fluid impulse, or the impulse of a vortex ring

static pressure, usually along the tube wall -+ -+

position vector: x - S

radius of a vortex ring to the core centerline (Fig. 2-5)

radius of a vortex ring to the outermost closed streamline (Fig. 2-5)

time

the thickness of a vortex ring (Fig. 2-5)

Eulerian flowfield velocity -+

z component of u

average piston speed

average jet or slug-flow velocity through the orifice

velocity of a vortex ring

non-dimensional ring velocity (Eq. (3-31»

volume of the vortex ring

volume of fluid ejected from the orifice

position vector from the origin

piston trajectory

piston stroke length

distance of the vortex ring from the orifice

axial coordinate (Fig. 2-3)

xii

r

e: -+ z;;

\!

p,e,z

p, z -p, z

Po

Pm 'T

p <P

'l' A A

<P, 'l' -+ w

i

o

GREEK SYMBOLS

total circulation of a fluid element, vortex filament, or vortex ring

R/po

position vector, designating a vortical fluid element or the dipole field (Fig. 2-1 or 2-4)

kinematic viscosity

cylindrical coordinates where z corresponds to the axis of the vortex ring

outer variables

inner variables

tube radius

mass density

p/po, z/po

p/R, z/R

piston stroke time (sec.)

velocity potential

axisymmetric stream function

non-dimensional potential and stream functions (Eq. (3-19))

vorticity vector

REPEATED SUBSCRIPTS

a quantity induced by the presence of boundaries

a quantity related to a vortex ring in an unbounded flow

DIMENSIONLESS GROUPS

CD coefficient of drag

Re Reynolds number

Rej,ReT jet and tube Reynolds numbers (Eqs. (4-2) and (4-3))

ReR,Rer ring Reynolds numbers based on Rand r (Eqs. (2-23))

xiii

A6telL Ylotiu.ng HebnhoUz' ~ acfm..Uutble. dMC.OVelL!I 06 the. laJAJ 06 volL.tex mouol1 -til a pelL6e.c..t £.-tquid--that ~, -t11 a 6luid pelL6e.c.ily de,6:t<;tu.te 06 V~C.OMty (OIL 6lLud 6'Uc.Uol1l-• • • tlLil, cU..6c.oveJtlj -tl1ev-ttably ~(Lgge,6U the. -tdea that HelmhoUz' ~ JUltg~ aile. the only .tJtue atOIM.

- LoJtd Kel.v-tl1, "011 VolL.tex Atom~"

1-1 HISTORICAL OUTLINE

1 l11troductio11

Vortex r~ngs have fascinated people for over a century. After

Helmholtz published h~s celebrated paper in 1867 describ~ng the basic

laws of the flow of fr~ct~onless, lncompress~ble fluids, a great deal

of excitement and act~vity developed around the observat~on that a

th~n core vortex r~ng ~n such a fluid would never change in character.

The permanence of the "Helmholtz ring" led Kelv~n [64, art~cle 1] to

propose the vortex r~ng as a model for the atom. D~fferences among

elements would be accounted for by d~fferent degrees of "knottedness"

of the vortex core, and the radiation of heat and l~ght by core

v~brations. Extens~ve mathematical analyses of "class~cal" vortex

rings continued through the turn of the century [10, 21-24, 62-64].

Serious exper~mental study of vortex r~ngs in a real fluid began

w~th the ~mpresslve flow visualization work of Krutzsch ~n 1939.

Further stud~es of vortex r~ngs were precluded by the onset of World

War II, until G.I. Taylor reklndled lnterest with a letter to the

editor of the Journal Qf Aoplled PhYS1CS ln 1953. Soon thereafter

J.S. Turner published his excellent paper on buoyant vortex rings with

the suggestion that such rings might prove useful ~n seed~ng clouds to

~nduce the productlon of rain. Several other papers appeared in the

fifties and slxties concerning buoyant vortex r~ngs [13, 41, 69, 70],

1

Sect10n 1_1 Introduct10n 2

along with the suggestloo that even 1f vortex r10gs would oot prove

useful 1n seeding clouds, perhaps they could be used to transport

10dustrial wastes hlgn 1nto the atmosphere. Fortunately for all of us

th1S scheme never bore fru1t.

Interest 1n vortex rings exploded 1n the sevent1es. What was

before a tr1ckle of papers soon became a steady stream and 1S now

show1ng slgns of becom1ng a flood. Deta11ed exper1mental stud1es of

neutrally buoyant vortex rings resumed once aga1n w1th publ1cat1ons by

Maxworthy in the West, and Osh1ma 1n the East. Mathematic1ans such as

Norbury, Fraenkel, and Saffman extended the theory of vortex r1ngs to

include larger cores and viscous effects. Extens1ve mathemat1cal

treatments of vortex r1ng stab111ty have come from W1dnall and her

group at M.I.T.

The rapld 1ncrease 1n 1nterest 10 vortex r1ngs is perhaps a

result of the1r increas10g appl1cation to a wide range of phys1cal

problems. Vort1ces are commonly found 1n nature at all scales, from

the flowf1eld produced by 1nsects, to l~rge atmospher1c low pressure

systems cover1ng thousands of square k110meters. Vortex r1ngs prov1de

an 1deal exper1mental and mathematlcal model for studY1ng general

aspects of vortex evolut10n. The rollup of shear layers into

vort1ces, the d1ffusion of vorticity from reg10ns where vorticlty 1S

concentrated, the convect10n of vort1c1ty and vortex f1laments, the

1nstabI1Ity and breakdown of vort1ces and vortex 11nes, the

development of turbulence from such breakdowns, vortex stretch1ng,

all of these tOP1CS relate to the study of vortex rings.

Vortex rings themselves have more d1rect applIcatIons to many

problems of practIcal and academ1c 1nterest. Current stud1es of the

larger scale structures in turbulent flows have led to 1nterest In

the vortex r1ng as a possible model for the large eddles observed

especlally 1n turbulent shear flows [9, 53]. In the development of

turbulent Jets, for example, vortex r1ngs can be observed to form at

Introduction 3

the lip of the Jet and to go through compllcated pairing processes to

form larger, less coherent rlngs farther downstream.* These vortex

rlngs are found to play an lmportant role as well in the structure of

the sound fleld around turbulent Jets [39]. Atmospherlc thermals can

be visualized as large, buoyant vortex rings [41, 59]. Birds such as

hawks and eagles soar around the center of these large vortlces,

uSlng the vertical winds and upward movement of the buoyant air to

gain altltude with 11ttle expenditure of the1r own energy [8].

Propulsive fl1ght of 1nsects and birds also involves the product10n of

vortex rings. In two recent articles [52], Rayner has modeled the

wake of blrds and 1nsects as chalns of vortex rings which carry away

the react10n momentum to the lift and thrust produced by the mot10n of

the w1ngs. Plasma phYS1C1StS have been 1nterested 1n the evolution of

vortex r1ngs as they travel through flu1ds such as as liquid helium II

[11, 12, 18,72,73], pr1nclpally to study the quant1zed nature of the

total clrculatlon. Vortex r1ngs were even used by by BJerknes in 1926

to model the occurance of sunspots!

In1tially, our interest 1n vortex rings developed from the study

of the trans1ent flowfield beh1nd arterial stenoses, or blockages 1n

the human arter1al system, caused by the buildup of cholesterol on the

arterial wall or from d1seased heart valves Wh1Ch will not open

completely. The pulsatlle nature of the blood as it 1S ejected

through these constr1ctions results in the production of rather

1ncoherent, turbulent vortex rings Wh1Ch travel up the artery and

eventually collide with the arterial wall. This has led, therefore,

to our interest 1n the effect of boundar1es such as a tube on vortex

ring evolution.

*E. Bouchard and w.e. Reynolds in the Department of Mechanical Engineering here at Stanford have produced beautiful movies of this process.

Sect10n 1-2 Introduct10n 4

1-2 OBJECTIVES AND ORGANIZATION

In the past, exper1mental stud1es have concentrated on vortex

r1ngs in unbounded flows, and over a l1m1ted range of Reynolds

numbers. Our emphas1s 1n this study is two-fold:

We would l1ke to exam1ne the dynamic processes and differences

1n characterist1cs from very low Reynolds number V1SCOUS vortex r1ngs

to very h1gh Reynolds number turbulent vortex r1ngs, and we would l1ke

to assess the effects of the wall as vortex r1ngs travel up a tube.

The dynam1c processes in which we are 1nterested 1nvolve the

loss of vort1c1ty from vortex r1ngs as they propagate. We would l1ke

to predict the change 1n strength, or total c1rculat10n as a funct10n

of time for vortex r1ngs at d1fferent Reynolds numbers. To accomp11sh

th1s dynamic calculation we develop a method based on a purely

k1nemat1c theory which allows us to relate the strength of a vortex

ring to quant1t1es wh1ch can be measured from flow visua11zat10n

experiments.

In the next chapter the background to th1s work is estab11shed,

and in Sect10n 2-5 the concepts beh1nd our method for calculat1ng the

strength of a vortex r1ng 1S d1scussed. In order to better understand

the theoret1cal and exper1mental analyses which follow 1n succeed1ng

chapters, 1t 1S suggested that Sect10n 2-5 not be sk1pped.

In Chapter 3 we develop the k1nematic theory for a vortex r1ng

in a tube. We find the potential and streamline f1elds for vortex

rings of different s1ze, and learn what the effect of the tube 1S on

the shape and speed of a vortex r1ng as compared w1th the same r1ng 1n

an unbounded flow.

Flow visua11zat10n experiments in which vortex r1ngs were

produced and observed as they travel up a tube are d1scussed 1n

Chapter 4. Qua11tat1ve observations and quantitative measurements of

vortex ring veloc1ty, s1ze, and shape are comb1ned to study

Section 1-2 Introduction 5

differences in vortex rIngs as they relate to Reynolds number.

In Chapter 5 the flow vIsualIzatIon measurements are combined

with the kinematic theory to calculate the total circulatIon as a

functIon of time for our experimentally produced vortex rings.

Relationships wIth Reynolds number are establIshed, and quantItative

dIfferences discussed, agaIn with emphasis on changes relating

prImarily to the Reynolds number of the vortex ring.

The maeiAtltcm! We knew tlutt at the ude the pent-up lA!ateM between the .u,lanc!J., 06 FeJlJlOf> and L0660den ILUJ.>h wdh -<.Nt~.u,ta.ble. v-tolenc.e, 60!!TntJ1g a wfuAepool 6JtOm w{uc.h no V~f>i!.l eveJr. ~c.ap~ It IAXI~ d~c./'..-tb-tl1g a f>p-t.-M.l, the. c.AJlc.um6eJr.enc.e 06 wluc.lt WM l~Mlt-tng by deglte~, and the boat ... 1AXUJ c.M/t-te.d along w.uft giddy f>peed ••• we heM.d It c.JtM (ung no.u, e, the boLt.6 gave WIlIj,

and the boat, toltn 6/tom ~ g~oove, WM hU!tled l-tke a .6tone 6/tOm a f>l-tng -tlttu the rn-tdJ.,t 06 the wl'U/tlpool • .

2-1 KINEMATICS OF VORTICITY

2 Backgrou11d

Vort1c1ty 1S a measure of the angular velocity, or spin of fluid

part1cles. In a flu1d of constant dens1ty, as we consider here, spin

is imparted by frictional forces 1n the flu1d, for example along a

boundary or surface within the flow. Although vortic1ty is a dynamic

quantity a great deal can be learned by consider1ng the kinematic

relat10nsh1ps between the vorticity field and the velocity field, that

is, by cons1der1ng only the conservation of mass and the definition

for vort1c1ty.

BIOT-SAVART LAW. Consider an incompress1ble and, for the

moment, an unbounded flow at rest at inf1n1ty. At the time t the -+ -+-+

veloc1ty at the point x 1S given by u(x) and the vorticity at the -+ -+-+

point ~ by w(s). Thus we have

Conservation of Mass:

Defin1t10n of Vort1city:

-+ div u = 0

-+ -+ curl u = w

(2-1 )

(2-2 )

-+ -+ We can satisfy Eq. (2-1) identically by letting ti = curl B. B is

-+ chosen such that div B = 0, so that application of Eq. (2-2) results

in the Poisson equat10n:

-+ -w

6

SectJ.On 2-1 Background 7

with the solution: 00 -+-+

= __ 1 __ fff w(~) d~3 4 'IT -00 r (2-3)

-+ -+ -+ 1-+1 As shown in F1g. 2-1, r = x - ~ and r = r. Taking the curl of Eq.

(2-3) results 1n the well known law of B10t and Savart:

-+-+ u(x)

1 4'IT

00 -+ ++ JJf r Xr~(~) d~3 _00

-+

(2-4)

At any t1me t the velocity at the point x is given by an 1ntegral of

the vortic1ty field over the whole flowf1eld. Two 1mportant features

of Eq. (2-4) should be g1ven spec1al note:

1. Th1s 1S a kinemat1c expreSS10nj only conservation of mass has been

used 1n its der1vation. Thus, even for a t1me-dependent flow,

laminar or turbulent, where vort1c1ty is d1ffused and convected,

if at any time the vort1c1ty field is known or can be well

approximated, the veloc1ty f1eld can in principle be computed from

Eq. (2-4).

2. The veloclty f1eld 1S related linearly to the vort1city f1eld.

The vortic1ty field can therefore be d1v1ded 1nto elements, the

veloc1ty f1eld found for each element separately, and summed to

find the total velocity at any point 1n space and t1me.

BOUNDARIES. Although Eq. (2-4) was derived for an unbounded

flow, 1t can be extended to a flow w1th boundar1es by add1ng a -+ -+

potent1al velocity (i.e., 1rrotational and divergenceless) u (x) to 1

Eq. (2-4), chosen to satisfy the boundary cond1tions.

VORTEX FILAMENTS. In vortical flows it 1S common to find

vort1city concentrated along curved lines, so that 1t 1S natural to

1deal1ze the concentrated reg10n of vortical flow as a vortex tube, or

1n the limit of zero tube rad1us, a vortex f1lament. The vorticity

Section 2-1

o

FIGURE 2-1

Definitions for a vortical flowfield

FIGURE 2-2

Definitions for a vortex tube or filament

Background 8

Rotational Fluid

Sect~on 2-1 Background 9

content of a vortex tube (or a flu~d element) is characterized by the

circulation, or strength r where

-+ -+ r = !J u • d1', -+ -+

== II w • dA (2-5) C A

C is a closed material curve around the vortex tube (or fluid element)

and A ~s the cross-sect~onal area, as shown ~n Fig. 2-2.

Mathemat~cally a vortex tube ~s often ideal~zed as a f~lament, or line

where the strength ~s g~ven in the l~mit A -+ O. The accuracy of such

an idealization depends on the degree to which vort~city ~s

concentrated, and the reg~on of the flow in wh~ch we are ~nterested.

It ~s often the case that "far" from the reg~on of concentrated

vort~c~ty the velocity f~eld ~s accurately descr~bed by vortex

filaments, whereas the flow "near" the vort~cal reg~on must take ~nto

account the deta~ls of the vort~c~ty d~str~but~on. -+

Us~ng the d~vergence theorem and the divergenceless nature of w

~t ~s easy to show that the c~rculat~on r must be constant along a

vortex tube. Th~s k~nemat~c result has the consequence that a vortex

tube cannot end ~n the flu~d. The f~laments must e~ther form loops

(e.g., vortex rings) or extend to the boundary (e.g., the horseshoe

vortex observed ~n turbulent boundary layers).

For a vortex filament of strength r the Biot-Savart Law, Eq.

(2-4), reduces to the more fam~l~ar form:

-+ -+ r u(x)= - 4n

-+ -+ r x d1',Z;

rJ r 3

filament -+ -+

where 1', Z; ~s along the filament ~n the dHection of w.

(2-6 )

IMPULSE. One add~t~onal kinemat~c express~on of ~nterest ~s

that for the flu~d ~mpulse, the total ~mpuls~ve force wh~ch would be

requ~red to set the vortical flowf~eld into motion. The total ~mpulse -+ P, der~ved by Batchelor [2, p. 518] and Lamb [29, p. 214] has the

Sectl.on 2-1

units of momentum and is g1ven by:

-+ p = ~ -+ -+-+ 3 fff Z;; x w(Z;;)dZ;;

where P 1S the mass density of the flu1d. m

2-2 DYNAMICS OF VORTICITY

Background 10

(2-7)

Like momentum, vorticity can diffuse down a gradient slope

through the act10n of v1scosity, or 1n the case of turbulent flows,

through the action of small scale turbulent mot1ons. The dynam1cs of

V1SCOUS diffus10n in an 1ncompress1ble fluid 1S described by the

vort1c1ty equat10n:

-+ 3w -+ -+ -+ -+ 2 -+ - + u • 'il w = w • 'il u + \) 'il w 3t

(2-8)

The left hand s1de is, of course, the total derivative accounting for

vortic1ty changes 1n flu1d elements as they are convected w1th the

flow. The term 1nvolv1ng \), the k1nematic v1scosity, describes -+ -+ -+

V1SCOUS d1ffus10n of vorticity, and w·'ilu gives the change 1n w due to

vortex stretching. If a flow 1S of suff1ciently h1gh Reynolds number

to disregard V1SCOUS d1ffus10n over the time per10d of interest, yet

low enough to d1sregard turbulent diffus10n, the vortical content of a

flu1d element convected w1th the flow would only change via vortex

stretch1ng.

KELVIN/HELMHOLTZ THEOREM. ApplY1ng the total der1vative to the

def1nltion of r 1n Eq. (2-5) and 1nsertlng the 1ncompress1ble Dti

momentum equatlon for Dt one arr1ves at the followlng expresslon for

the c1rculatlon of a fluld element as it is convected with the flow:

Dr Dt

-\) ~ curl ~ • d1 = \) II 'il2~ • dA C A

(2-9)

Sect10n 2-2 Background 11

where C and A are defined as in Eq. (2-5). The circulat10n can change

only by the d1ffusion of vort1city from the element. If we aga1n have

a flow 1n wh1ch we can neglect the effects of fr1ct10n we arrive at

the Kelvin/Helmholtz theorem:

Dr Dt

o (inviscid) (2-10)

wh1ch states that the c1rculat10n of every flu1d element rema1ns

constant 1n time. Th1S has the 1mportant consequences that vortex

tubes are convected w1th flow at the local flu1d veloc1ty, and that a

vortex tube is always composed of the same flu1d part1cles. The

previous considerat10ns of the vort1c1ty equat10n show that as a

vortex tube is stretched by the local veloc1ty f1eld the magnitude of

the vort1c1ty w1th1n the tube 1ncreases. S1nce r must rema1n

constant, the area of the tube must therefore decrease.

In such flows where d1ffus10n of vort1c1ty can be neglected, the

B10t-Savart law can be used to descr1be both the dynamic as well as

the k1nematic behav10r of the flow. The veloc1ty f1eld is "1nduced"

by the vorticity f1eld. Since each vortex element moves with the

local fluid velocity, the self-1nduced velocity f1eld found from Eq.

(2-6) can be 1n 1ntegrated 1n t1me to f1nd the subsequent location of

the vortex elements. Aga1n uS1ng Eq. (2-6) the veloc1ty f1eld at that

later time can then be calculated and the process repeated.

In general, of course, diffusion of vort1city due to v1scosity

or turbulent motions is always present, but in many flows contain1ng

concentrated regions of vort1city, the rate of d1ffusion 1S slow as

compared w1th the convect10n of vort1c1ty, and 1mportant features of

the flow can be understood from the 1nv1sc1d cons1derat10ns above. An

example is the instab11ity and rollup of a shear layer lead1ng to the

formation of 11near vort1ces, wh1ch then proceed to pa1r [78].

INVARIANCE OF IMPULSE. ApplY1ng the total derivative to Eq.

(2-7) and using the vorticity equation, one can eventually arrive at

Section 2-2 Background 12

an expression for the var1ance 1n the total fluid 1mpulse 1n a bounded

flow. If the vortic1ty and its normal derivat1ve are zero along the

boundary the result is: -+

dP dt = ~

-+

P ¢:it u 2 dS m

S

(2-11a)

where S is the boundary surface. For an unbounded flow (S -+ 00) at

rest at infinity we see that

-+ dP dt o (unbounded) (2-11b)

Thus the total flu1d 1mpulse of an unbounded fluid is invariant for

both V1SCOUS and 1nvisc1d flows.

2-3 VORTEX RINGS IN AN IDEAL FLUID

2-3.1 The Classical Vortex Ring

A vortex r1ng can be defined as a vortical region of fluid where

the mean vorticity field has an azimuthal sense and is concentrated

(peaks) along a c1rcle.* It 1S natural, therefore, to model the

vortex ring as a c1rcular vortex tube, where vort1city is conf1ned to

a torus of small cross-section. Helmholtz [20] proposed such a model

for a vortex r1ng in an ideal (incompress1ble, 1nvisc1d), unbounded

flu1d about 100 years ago, and its propert1es were extensively studied

around the turn of the century [10, 21_23, 62-64].

NON-STEADY COORDINATE SYSTEM. Consider an axisymmetric flow

without swirl. We define a cylindrical coordinate system (p, 6, z)

*We are concerned here with circular vortex rings, although more general shapes such as elliptical vortex rings can be found in nature and have been studied as well (1, 14, 26).

Section 2-3.1 Background 13

flxed ln space* such that at the tlme t a torus of vorticity of radlus

R is located in the z = 0 plane axisymmetrically placed about the

z-axis (see Fig. 2-3). The torus, called the vortex core, has a

clrcular cross sectlon with radlus ~ and a total clrculation, or

strength r. Outside the core the flow lS taken to be lrrotational.

Due to its self-lnduced veloclty fleld the vortex rlng lS moving wlth

speed U in the +z dlrection, so we are deplcting the vortex rlng ln a

non-steady frame of reference.

CIRCULAR VORTEX FILAMENTS. The classlcal vortex rlng is a thln

torus of vortlcity embedded ln a potentlal flow, as sho~~ in Flg. 2-3.

By "thln" we mean ln the llmit aiR -+ O. In this llmit the flowfleld

outslde of the core is that for a clrcular vortex filament of strength

r (the flowfield wlthln the core, of course, depends on the

distributlon of vorticity). In this region the flow is lrrotatl0nal,

so we can define a veloclty potentlal, ~:

ti = V ~ (2-12)

ApplYlng Stokes Theorem to the Biot-Savart law, EQ. (2-6), one can

obtain the following representalon for the potential field due to a

closed vortex fllament:

~(;D = (2-l3)

As indlcated ln Flg. 2-4, A lS the surface bounded by the vortex -+ fllament-loop, ~ 1S a pos1t10n vector to a point on that surface, n 1S

the outward normal to the surface (defined by the d1rect10n of r), and -+ -+ -+ r = x - ~ . Eq. (2-13) shows that mathematically a vortex

*By "fixed in space" we mean fixed with respect to the stagnant flow at infinity or, in the case of the vortex ring in a tube, fixed with respect to the boundaries.

Section 2-3.1

FIGURE 2-3

The cylindrical coordinate system

o

FIGURE 2-4

A vortex filament-loop

+ x

Background 14

filament

p


filament-loop of strength r can be replaced by a uniform d~stribution

of dipoles of strength r over the surface bounded by the loop. Th~s

result applies to any arb~trar~ly shaped vortex filament-loop. In

particular, a circular vortex f~lament can be replaced by a disk of

d~poles of un~form strength; the potential at any po~nt ~s Just the

~ntegral of these d~poles over the d~sk. Th~s suggests, as we show

expl~c~tly in Sect~on 3-1.1, that a vortex r~ng in the non-steady

frame of reference produces an outer flowf~eld w~th a basically dipole

character. From Eq. (2-7) the flu~d ~mpu1se for a circular vortex

f~lament of strength r and rad~us R is found to be

-+ where P = pz.

(2- 1 4)

VELOCITY. As d~scussed in the prev~ous sect~on, in an ~dea1

f1u~d a c~rcular vortex tube w~ll move due to ~ts self-~nduced

velocity field. Kelv~n [64] and Lamb (29], us~ng the Biot-Savart law,

found the velocity of the class~cal vortex ring w~th a constant

vorticity distribut~on ~n the core (i.e., the core rotates as a sol~d

body) to be:

r [8R a ] U = - Q,n - - 1t; + 0(-) 4nR a R

(2-15)

Th~s formula has been extended by Saffman [54], Fraenkel [15], and

W~dnall [14] to an arb~trary distr~but~on of vort~c~ty in the core:

r u = 4nR (2-16)

where A is a constant wh~ch depends on the vort~city d~stribut~on.

Saffman presents A ~n the following form:

A = a s f (2n f r o 0

s'wo (s')ds,)2 ds s

s ~s a radial coordinate from the core's center and woes) is the f~rst

Sect10n 2-3. 1 Background 16

order vorticity distribut10n. For a uniform d1stribution A 1S 1/4.

For the cont1nuous d1str1but10n woes) = (s2_a2)2 which has been used

by W1dnall in her studies of the stability of vortex rings, and 1S

more real1stic with respect to real vortex rings l36], A = 5/6 so

that:

u ~ r

4nR ( in 8R _ l )

a 3 (2-17)

Note the presence of a logarithmic s1ngular1ty as aiR + O. A result

of locally induced velocities which in the limit of a vortex filament

become 1nfin1te, a logar1thm1c singularity in veloc1ty 1S a general

feature of a curved vortex filament.

In an ideal flu1d r is constant, and S1nce vort1city cannot

d1ffuse from the core, A cannot change. For an unbounded flow the

1mpulse 1S 1nvar1ant as well, so from Eq. (2- 1 4) the radius of the

vortex r1ng must rema1n constant. Thus, a thin core vortex r1ng in an

unbounded, 1ncompressible, completely fr1ctionless flow would travel

to inf1n1ty w1th all properties invar1ant.

MOVING FRAME OF REFERENCE. In a coordinate system f1xed in

space, the 1nstantaneous streamline field resembles that of a point

dipole. If we allow the coord1nate system as def1ned 1n F1g. 2-3 to

move with the vortex r1ng,* however, three regions are def1ned:

1. The tor01dal core region in which the rotational flu1d is

conf1ned.

2. A spheroidal reg10n surrounding the core defined by an outermost

closed streamline with a front and rear stagnation point.

3. The outermost region w1th streaml1nes which extend to 1nfinity.

A classical vortex r1ng 1n steady and non-steady coordinates is

*Such a frame of reference need not in general be steady (i.e., moving with constant speed), but for a vortex ring in an ideal fluid it is.

Section 2-3.1

R'

moving with the vortex ring (steady)

FIGURE 2-5

Background 17

fixed in snace (non-steady)

The classical vortex ring in steady and non-steady coordinates.


compared 1n F1g. 2-5. The f1u1d w1th1n the sphero1da1 region 1S

conf1.ned to tnis rE'g1.on and 1.S transported along w1.th the core. When

uS1.ng the term, "vortex r1ng," we refer to the whole spher01dal region

of fluid, including the region 1n Wh1Ch the vortic1ty is

concentrated-- the vortex core.

2-3.2 Thick Core Vortex Rings

The class1cal vortex ring is a torus of vortic1ty in the lim1t

of vanishing core thickness. We consider now vortex rings w1th thick

cores. For such rings the cross-sectional area of the core is not

circular and the d1str1bution of vortic1ty within the core must be

cons1dered.

For an ideal flu1d the vorticity equat10n, Eq. (2-8), can be

wr1tten for axisymmetr1c flow w1thout sW1rl as

-+- " where w = w e w

neglected 'p

where 'l! 1S the

D (!E) =0 Dt P

(2-18)

That is, for a flow where viscous d1ffusion can be

must be constant along a flu1.d streamline. Thus

w f('l!) (2-19) P

stream function in cylindr1.cal coordinates:

1 a'l! 1 a'l! (2-20) u az u ap P P z P

The stream function 1S defined so as to satisfy cont1nuity; applying

the def1.n1t10n for vorticity, then, results in a k1nematic equation

wh1.ch can be solved for a g1ven vort1city d1str1bution to find If'.

When comb1ned w1th the dynamical result in (2- 1 9) the stream function

for a th1ck core vortex ring must sat1sfy:

(2-21)

o elsewhere


Several mathematic~ans have sought solut~ons to (2-21) for the

case where f(~) is a constant. At the extreme of th~ck core vortex

r~ngs ~s the solution found by Hill [24] for a spherical vortex 1n

wh~ch vort~city occup~es a sphere such that f(~) = C, or equ~valently,

w = C p. An exact solut~on to Eq. (2-21) can be found for th~s case.

The stream function and veloc~ty for H~ll's spher~cal vortex are g~ven

by:

u

3 r [1 _ ( XR

, ) 2 ] 2" STIR'

r 2 STIR' r

(2-22)

where R' ~s the sphere radius and x 2 = p2+ Z2. We note that even ~n

the extreme case of a spherical vortex the same bas~c relations among

the velocity, strength, and radius pers~st. That is, for both th~n

and th~ck core vortex rings the r~ng speed ~s d~rectly proportional to

the total c~rculat~on and ~nversely proportional to the rad~us.

Norbury [42] has numer~cally computed solut~ons to Eq. (2-21) for f(~)

constant, and has shown that the class~cal vortex r~ng and H~ll's

spher~cal vortex are two extremes of a fam~ly of vortex rings based on

the s~ngle parametera~aeff/R where aeff ~s the rad~us of the core as

~f ~ t were circular w~ th the same area. a. = 12 represents H~ll' s

sphericnl vortex and a. -r 0 ~s the class~cal vortex r~ng of Kelv~n and

Lamb. The computed solut~ons ~nd~cate that for a. ~ 0.2 the

streaml~nes with~n the core and the core cross-sect~on are very nearly

circular.

2-4 VORTEX RINGS IN A REAL FLUID

2-4.1 Vortex R~ng Formation

Vortex rings are typ~cally formed by ejecting a flu~d through an

or~fice or from a tube. As the flu~d is eJected, a cyl~ndrical sheet

Sectl0n 2-4.1 Background 20

of vortlclty emerges, and due to ltS self-induced veloclty fleld

(Section 2-2) beglns to spiral, concentrating vorticlty in a toroldal

regl0n.* As the rollup of the cyllndrlcal vortex sheet continues, the

vortlclty ln this regl0n lncreases, until the vortex rlng breaks from

the orlflce and beglns movlng lnto the amblent fluid with lmpulse and

strength determlned by the magnltude and amount of vortlcity

entrained.

When the eJected fluid lS dyed the process of formation and the

extent of the vortex rlng can be vlsualized. A well-formed vortex

rlng lS generally observed to consist of a darkly dyed torus

surrounded by a more lightly dyed spherold of fluid. The more darkly

dyed region marks the core of the vortex rlng. Measurements by

Sulllvan ~~ [60J and Maxworthy [36J show that even for relatlvely

"thick" vortex rings the vorticlty dlstributl0n lS hlghly peaked ln

the core with smaller amounts of vortlclty resldlng ln the outer

region. In Flg. 2-6 are sketched the velocity and vortlclty

dlstr1butl0ns measured by Sullivan ~ ..a.l..... for a "thin" core and

"thlCk" core vortex rlng (aiR = 0.075 and 0.27 respect1vely). The

core d1ameter, 2a, 1S deflned as the d1stance between the maX1mum and

mlnlmum ln veloc1ty.

2-4.2 Vortex Ring Evolutl0n

The character1stlcs of vortex rlngs as they evolve are dependent

to some extent on the lnlt1al state of the vortex ring, WhlCh depends

on the condltlons at format10n. The rlng strength, for example,

inlt1ally depends on the magnitude and amount of vorticity ejected,

WhlCh ln turn lS related to the velocity at eJection and volume of

eJected fluld WhlCh forms the vortex ring (see Sectlon 5-2.2). The

*For beautiful pictures of vortex ring formation see Batchelor (2, plate 20) and Magarvey, MacLatchy (32).


(a) (b)

a - = R

0.075, ReR= 18710 a - = R

0.27, R~= 3040

sketched streamlines

I I I I

~

FIGURE 2-6

2

-2

U

u-U U

____ I

I I I I I I

-----vorticity distribution

The velocity and vorticity distributions in a real vortex ring.

From Sullivan, Widna11, and Ezekiel (1973).


s~ze of the vortex ring ~s expected to be related in some way to the

diameter of the or~f~ce through which the flu~d ~s ejected [28], and

the development of turbulence to the roughness and irregularities in

the orifice geometry.

In general, as the vortex r~ng propagates through the ambient

flowfield, vorticity d~ffuses from the core region and small amounts

of ambient fluid are entra~ned ~nto the rear of the vortex ring,

mix~ng with the flu~d exter~or to the core [32]. Vortic~ty ~s

subsequently convected ~nto a wake region along with some dye,

result~ng ~n a more lightly colored outer reg~on ~n comparison with

the core. The loss of vort~c~ty from the vortex r~ng results in a

decrease in r~ng strength and impulse· and, since U and r are directly

related, a decrease in velocity.

RING REYNOLDS NUMBER. The relative ~mportance of ~nertial to

viscous forces in the evolut~on of the vortex r~ng is characterized by

the Reynolds number. Two common definlt~ons for the r~ng Reynolds

number are based on the r~ng radius and velocity (ReR) and on the

total c~rculat~on (Re r):

Re = R 2RU v

r (2-23)

Since r ~ RU, Re R and Rer are expected to be closely related (see

Section 5-3).

VISCOUS VORTEX RINGS. At low Reynolds numbers (ReR ~ 600-1000)

vortex ring evolut~on is governed by v~scous d~ffusion of vorticity

from the core and ~ts subsequent convect~on into a laminar wake. As

~llustrated ~n Fig. 2-7a streaml~ne surfaces of viscous rings remain

smooth and the core tends to be th1ck and visually not well defined.

*The impulse of the ring plus wake is conserved by Eq. (2-llb).

Section 2-4.2

core

viscous wake

(b) Turbulent Vortex Ring

FIGURE 2-7

U

,q I I \ , I \

~ t ~

Laminar and turbulent vortex rings.

Background 23

(a) Viscous Vortex Ring

turbulent ~_- mixing

entrainment

turbulent wake


Tung and T~ng [66] and Saffman [54] have derived an express~on

for the velocity of a thin core of viscous, rotational flu~d embedded

~n a potent~al flow:

U = r [,\1,n 8R - 0.558 + 0 UVt_ ,\1,n Vt ) ] (2-24) 41TR 14Vt / R2 RT

Apparently, for th~n core rings (i.e., ~n the l~mit aiR ~ 0) the core

rad~us grows due to v~scous diffusion like M and the velocity

decreases as -In(vt). Accompanying the growth of the core there ~s an

exponentially slow decrease in r~ng strength [54], and since the

impulse must rema~n constant, from Eq. (2-14) there ~s a slight

~ncrease in R.

For asymptotically large t~mes, when t'V R (Le., a/R'V 1), one

f~nds a much more rapid decrease in U and increase ~n R [25, 52]:

r 'V (v t) -1/2 (2-25)

V~scous vortex rings ~n a real flu~d generally have th~ck cores,

and measurements show a rate of decrease in velocity and increase in

radius somewhere between that pred~cted by the extreme thin and th~ck

core models Just ment~oned (see Sect~on 4-4), although measurements

made for large times [27] do suggest that the velocity approaches the

asymptotic lim~t g~ven in (2-25).

INSTABILITIES. Many researchers [28, 30, 34, 48, 73] have found

that as the Reynolds number increases, although the flow is st~ll

laminar, azimuthal waves begin to appear along the core of the vortex

ring. F~ve waves are typically observed to develop at ring Reynolds

numbers on the order of 600 to 1000; as the Reynolds number increases

the number of waves ~ncreases. At the lower Reynolds numbers unstable

waves are observed to grow unt~l the vortex ring collapses, but for

higher Reynolds numbers the growth of unstable waves leads to a

"break~ng" process and the emergence of what is generally described as

a more stable vortex ring.


The instab111ty of vortex rings has been stud1ed mathemat1cally

by W1dnall ~ ~ [74-77, 66] and Saffman [56]. Perturbations are

found to become unstable when the locally induced rotat1on of waves

around a circular vortex filament 1S forced to zero by the veloc1ty

field 1nduced by the rest of the vortex r1ng.

TURBULENT VORTEX RINGS. As the Reynolds number lncreases

further, the wave number of the unstable waves increases and small

scale turbulence begins to develop. The Reynolds number at WhlCh

turbulent motions begin to dominate the dynam1cs depends to a large

extent on 1rregular1t1es and roughness in the or1flce geometry, the

method of ejectlon [57], the state of the amblent fluid, etc., but

transltional Reynolds numbers on the order of 20000 to 30000 are

typical for carefully produced vortex rings. Because of turbulent

mixing, the outer region of dyed flu1d becomes lighter 1n color,

revea11ng a fa1rly well-deflned core. The core to ring rad1us of

vortex r1ngs tends to decrease as the Reynolds number 1ncreases [60,

57, 36], and the small scale motlons ln the turbulent core are

relatlvely imperv10us to the larger scale motions 1n the outer m1xlng

reg~on [35, 68]. As ~nd~cated by Fig. 2-ib, turbulent diffus~on and

convection of vorticity lnto a turbulent wake results in a rapid

decrease in ring strength and veloc1ty.

2-5 PRELIMINARY CONCEPTS BEHIND THE PRESENT STUDY

Fig. 2-8 outlines the range of ring Reynolds numbers WhlCh have

been covered in experimental studles of vortex r1ngs, lncluding the

present study. In general, researchers have concentrated on a limlted

range of Reynolds numbers. The transition in characterlstlcs from low

Reynolds number, V1SCOUS vortex rings to high Reynolds number,

turbulent vortex rings has not been adequately stud1ed. This is an

important aspect of the present work:

t,:j 100 H 0 c:::: :;0 t%j

N I

CO

(') 0

~ P-l Ii f-'. CIl 0 ::s 0 I-h

:;0 CO

~ 0 ~ 0.. CIl

::s § C" CO Ii

. ~ Ii OSHIMA (1972) P-l ::s

OQ CO

~ f-'. rt ::r'

't:I Ii CO ~ f-'. 0 ~ CIl

CIl rt ~ 0.. f-'. CO CIl

Ring Reynolds Number,

500 1000

• • MAXWORTHY (1972)

...

2RU -\)

5000 10000

I IIII ~ .. KRUTZSCH (1939)

:~

.. MAXWORTHY f1972) 1.

1 MAXWORTV,Y <1974 . .,.

50000

SALLET and WIDMAYER (1974) LEISS and DIDDEN-(1976

f t ~uLLlr'IWlliLr tfrEL (19T . ~

IIII I PRE,ENT1 SLY' IJJ I _ I J I

tI.l co n rt f-'o o ::s N I

\J1

OJ P-l n :;>;"

OQ Ii o § 0..

N 0'\


To study the changes in the structure of vortex rings and the dynam1c

processes which control the1r evolution, from very low Reynolds

numbers where v1scosity dominates, to very h1gh Reynolds numbers where

turbulent diffusion dom1nates.

The dynamic processes 1n wh1ch we are interested 1nvolve the

loss of strength and impulse from vortex rings as they evolve,

processes wh1ch relate to the d1ffusion and convect10n of vort1c1ty

out of the vortex r1ng 1nto a wake. A measurement of the strength (or

total circulation) of a vortex r1ng at even one point 1n space,

however, 1S difficult to obtain. In the past researchers have used

laser doppler veloc1meters [50, 36] and hot W1re anemometers [57] to

measure the velocity around the core of a vortex r1ng in order to

compute the total circulation. Such measurements rely on a repeatable

production of the vortex r1ngs, and the use of hot wires raises

quest10ns as to 1nterference with the inner flowfield. Furthermore,

t1me constraints have allowed only a very limited number of such

measurements.

BASIC APPROACH. We propose as part of the present study a

method for computlng the total clrculatlon of a vortex ring from

relatively slmple flow visuallzation measurements of the size, shape,

and velocity of the vortex rlng. The basic ldea is as follows:

USlng the B10t-Savart law we find the stream functlon in

non-steady coordinates for a vortex ring of strength r and radius R

aXlsymmetrlcally placed in tube of radlus Po (results for an unbounded

r1ng are obtained 1n the limlt RI Po -+ 0). Transferr1ng to a

coord1nate system mov1ng w1th the vortex ring at the speed U, the

outermost closed streamllne outllnes the volume of the vortex rlng.

Defin1ng the thickness, T, and outer radius, H' of the vortex r1ng

(see Flg. 2-5), we compute the kinematic relationships among the

parameters R/Po , r, U, and T/H'. A measurement of any three of

these quantlties will therefore yield the fourth. In particular, from


flow visual~zation measurements of R/po , U, and T/R' at the time, t,

we can compute r(t), the total circulation of the vortex ring at that

time.

Let us consider this approach in more detail.

KINEMATICS. Consider again the kinematic nature of the

Biot-Savart law, Eq. (2-4). As discussed ~n Section 2-1, at any given

time, whether the flow is laminar, turbulent, viscous, or inv~scid, if

we know the vorticity field we can in princ~ple compute the velocity

field from Eq. (2-4). In general we do not know the vorticity f~eld

exactly, or the integrat~on of Eq. (2-4) ~s exceedingly difficult.

However, in many cases the flowfield can be well approximated and the

integration simplif~ed by d~vidng it into vortex elements and modeling

each element as a filament. In this way the simpler integration of

Eq. (2-6) can be used. Because the Biot-Savart law is linear in

vorticity, the individual flowfields of the vortex elements can then

be summed to find the total flowfield. This ~s the case with vortex

rings. We divide the vorticity f~eld into two parts: the vortex ring

itself, and the wake.

APPROXIMATIONS. Furthermore, we make the following

approximations:

1 Since vort~city is highly concentrated in the core (see Fig. 2-6)

we anticipate an inner and outer reg~on to the flowfield.* The

inner region (~n and "near" the core) will depend on the

distr~bution and extent of vorticity w~th~n the core. The

flowf~eld in the outer reg~on ("far" from the core), however,

should depend only on the total c~rculation of the vortex r~ng,

*Such observations have led to the use of Matched Inner and Outer Expansions by Tung and Ting (67) to study a thin core viscous vortex ring.

Sechon 2-5 Background 29

and not the details of the distr1bution of that vort1city. The

extent of the inner and outer regions will, of course, depend on

the degree to which vort1city 1S concentrated in the core, however

we assume that for tYP1cal vortex rings the vort1c1ty is

suff1ciently concentrated that the shape of the vortex ring can be

accurately predicted by concentrating the total circulat~on of the

vortex r~ng along a c~rcular vortex f11ament.

2. The amount of circulation lost to the wake 1S expected to be small

~n comparison with the total c1rculat1on of a vortex ring. We

assume, therefore, that the circulation per unit length behind

typ~cal vortex r~ngs ~s such a small fraction of the total

c~rculat~on with~n the r1ng that the vorticity 1n the wake can be

neglected in computat~ons of the shape of the vortex r1ng.

MODEL. The accuracy of these two approx~mations will, of

course, have to be tested (see Sections 3-3.2 and 5-2.1), but for the

purposes of relat1ng the total c1rculat1on to the r1ng s1ze, shape,

and velocity, we can now model the vortex ring as a circular vortex

f11ament axisymmetrically placed in an infin1tely long tube. Using

the known solution for the potential function of a circular vortex

f11ament in an unbounded flow, we derive the induced potential which

must be added to Eq. (2- 1 3) to account for the tube walls. From this

solut1on the stream funct10n is deduced and the stream11nes in

non-steady coord1nates calculated. Transferr1ng to a coord1nate

system moving w1th the vortex r1ng, we assess the effect of the tube

on the s1ze, shape and velocity of the vortex r1ng, and compute the

relationships among the non-dimensional parameters

T R and U R Po, r/41TR

(2-26a)

and among R' R and U R , Po, r/41TR

(2-26b)


Because the details of the core are often washed out, flow

visualization measurements of HI are generally more accurate than of

H. By combining the computed relationsh~ps among the parameters in

(2-26a) and (2-26b) we find that either H QC RI can be used to compute

the ring strength. Thus, from flow v~sualizat~on measurements of

HI(t), T(t) and U(t) we can compute both rand H as a funct~on of

time.

It should be po~nted out that the approximation of concentrating

the strength of the vortex ring along a circular vortex filament in

order to compute T and HI is not the same as that made by Kelv~n and

Lamb to compute the velocity of a thin core vortex r~ng, Eqs. (2-15)

and (2-16). In calculating the velocity, dynamics enters through the

use of the Kelvin/Helmholtz theorem for an ~nviscid fluid, Eq. (2-10).

Under this approx~mat~on the vortex core moves with the local fluid

velocity, which ~n the lim~t a/H + 0 becomes infinite. The

approximation used in our calculat~ons assumes that the shape of the

vortex ring ~s not much affected by the finite size of the core, so

that for such a calculation the strength of the vortex ring can be

concentrated along a c~rcle. The calculation is totally kinematic,

and s~ngularit~es do not appear. Furthermore, k~nemat~c relationships

are very general ~n that they apply at any instant in t~me where the

model ~s suffic~ently accurate, and for both laminar and turbulent

flows. Of course, dynam~cs must be included in some way to predict

anyone of these quantit~es; however, by combining the kinemat~c

theory w~th relatively s~mple experiments, ~mportant dynamic

information can be deduced.

In the next chapter we develop the theory and calculate the

relationships among the parameters in (2-26). From the theory we

assess the e~~ect o~ the tube on the size, shape, and velocity o~ the

vortex ring. In Chapter 4 flow visualizat~on experiments are

described, and measurements for vortex rings propagating up a tube are

discussed. Finally, ~n Chapter 5 the flow visualization measurements

are combined with theory to compute the total circulation as a


function of t~me for 17 vortex r~ngs with ~nit~al Reynolds numbers

ranging from 690 to 50100.

TIr.i6 plUmMY ma.t.teIL, 6oJ[·c.ed,into a. c.eJr.tain quantLty 06 motion cU.v.<.nely be6towed, 6a..U& .<.nto a. ~eJL.<.e6 06 wfWr.£.poo~ 011. vott.t<.c.e6, -<.n wfU.c.h the v~'<'b.e.e bocU.e6 6uc.h M p£.a.neu a.nd teNLe6:tJUai ob j ec.u Me c.aJrJr..<.ed Mound OIL .i.mpeUed towaJtd c.CJLtcu/t c.enbt.a.£. po-<.nU by the .e.a.w~ 06 vott.t<.c.a..e. motion.

-E.A. BUILtt on Ve6c.a4te6

3 Theory

3-1 INTEGRAL SOLUTION FOR A CIRCULAR VORTEX FILAMENT IN A TUBE

3- 1 .1 Harmonic Expansion for the Unbounded Ring

Cons1der the problem 1llustrated in Fig. 3- 1 • We seek the

solution for a c1rcular vortex filament with strength r and radius R

axisymmetr1cally placed 1n an 1nfin1te, rig1d tube of radius Po. The

flow 1S 1ncompressible, but not necessar1ly inv1sc1d. Because the

flow 1S axisymmetric (and without swirl), we introduce the cylindrical

coord1nate system (p, 8, z) fixed w1th respect to the tube such that

at the t1me t the vortex filament 1S located in the z = 0 plane. The

vortex ring 1S mov1ng 1n the +z d1rect1on w1th the speed, U; we seek

the Solut1on for the flowfield at an instant in time in a non-steady

frame of reference.

Outside of the vortex filament the flow is 1rrotational, so we

define the veloc1ty potential:

<3-1 )

-+ where x is a position vector to the point (p, z). ~ is further

div1ded into two parts:

~o + ~. 1

32

(3-2)

Section 3-1.1

FIGURF 3-1

z

Po

~-.~-:

I I I I

~U I I

Theory

P

Definitions for a circular vortex filament axisymmetrically placed in a tube.

z cp =

J;...,.:..:....~-=-~:...:...;\------C> p cp O,p > R

r

r = 2

FIGURE 3-2

Representing a circular vortex filament as a disk of dipoles of uniform strength.

33

Sect10n 3-1.1 Theory 34

~

¢o(x) 1S the potent1al for a circular vortex filament in an unbounded ~

flow and ¢i(x) represents the potent1al fleld 1nduced by the presence

of the tube. ¢. must satlsfy Laplace's equation and, together with 1

¢o, the boundary cond1tion of no flow through the tube wall:

(3-3a)

at P Po (3-3b)

¢o is given by Eq. (2-13) as:

-+ ¢o(x) = a r -+

II az (47Tr) dA(I;;) A

(3-4a)

where A lS the area of the disk bounded by the filament, ~ is a -+ -+-+

posltlon vector to a pOlnt on the disk and r = x-t;. (see Fig. 2-4).

ThlS express10n shows that the c1rcular vortex fllament can be

represented mathematically as a dlSk of d1poles with un1form strength

r. At the d1Sk surface there is a d1scontinuity in potential of

magnltude r as lllustrated ln F1g. 3-2. Lamb [29, p. 239] presents

the potential for such a disk of dipoles ln terms of Bessel functlons

of the flrst kind:

¢o (p, z)

co rR -2 I

o e-kz Jo (kp) Jl(kR) dk (3-4b)

Eqs. (3-4a) and (3-4b) are equivalent expressions for the non-steady

potential fleld due to a circular vortex fllament in an unbounded

flow.

In order to obtain the solution of the potenhal ¢. which must 1

be added to Eq. <3-4) to satlsfy the boundary condltlon at Po, we

expand 1nto an harmon1c series valld in the reglon p> R and find the

induced potential for each term separately. Conslder the outer

Sectwn 3-1.1 Theory 35

var~ables defined as

=£ - Z x p Z = - x = -

Po Po Po (3-5a)

and the inner var~ables:

- =£. Z x P Z = - x=-

R R R (3-5b)

where x = /p2+Z2. Rewriting <3-4b) m outer vanables we obtain

(3-6)

- ,..., where € = R/Po < 1. The ~nner var~able, P = P / € indicates that to

obta~n an expansion val~d in the far f~eld we should expand ~n the

lim~t € ..... O. Therefore we expand J 1 (k€):

Jl(ks) = (ks) (kS)3+ (kS)5 (3-7) -2- - zr:4 zr:42 ·6

Subst~tut~ng Eq. (3-7) ~nto (3-6) and noting that

results in

-kz e (3-8)

00

f e-kzJo(kP)dk o

(3-9)

The ~ntegral ~s the Laplace transform of Jo and has the value 1/X.

Thus ~n the reg~on P > R we have reduced the ~ntegral ~n Eq. (3-6) to

a ser~es of harmonic poles at the or~gin:

... ) (3-10)

The first harmon~c, wh~ch is dom~nant in the far f~eld, represents a

dipole field, show~ng that at infin~ty a vortex ring looks like a

po~nt d~pole. The h~gher order poles account for the var~ation from

Section 3-1 . 1 Theory 36

that of a purely d1polar f1eld due to the f1n1te Slze of the vortex

r1ng.

We note that the expanS10n given 1n Eq. (3-9) could have been

obta1ned as well from the representat10n for qio given by Eq. <3-4a)

by expanding 1/r 1n a Taylor ser1es about 1/x and carrying out the

integrat10ns term by term.

3-1.2 Monopole Solution: The Green Funct10n

Consider an expansion In a form slmilar to Eq. (3-10) for the

total potentlal, qi = qio + qi 1 :

¢(-p,z) = £r(£~ ~ £3 d3

<p + £5 ~ <p 2 2 dZ ~m - 22 '4 dZ 3 m 22 .42 .6 dZ 5 m - ... ) (3-11)

Exam1nation of Eqs. <3-3) and (3-10) shows that ¢m is the potentlal

funct10n for a monopole at the or1g1n 1n a tube. That lS, <p 1S the m Green funct10n to Laplace's equat10n w1th the Neumann boundary

cond1t10n and must sat1sfy: +

- 4mS (x)

= 0 at p 1

In a form which satlsfies Eq. <3- 12a) we wnte <p as: m 00

<Pm(p,Z) = ~ + { f(k)Io(kp) cos(kz)dk

(3-12a)

(3-12b)

(3-13)

where f(k) 1S chosen to satisfy (3-12b). 1/x has the 1ntegral

representatlon [6]:

00

~ = 1 f Ko(kP) cos(kz)dk x 'IT 0

I and K are mod1f1ed Bessel functlons of the first and second k1nd

SectJ.on 3-1.2 Theory 37

respectlvely. So Eq. (3-13) can be wrltten: (X)

~ = J [~~o(kP) + f(k) Io(kP)] cos(kz)dk m 0 7T

Applying the boundary condltion, (3-12b) requlres that

f(k)

Insertlng thlS expreSSlon into (3-13), the Green functlon becomes:

(X)

~m(p,z) =! + ~ J KI(k) Io(kp) cos(kz)dk x 7T 0 I I (k)

(3-14)

3-1.3 The Potential and Stream Functions In Non-Steady

Coordinates

Substltuting Eq. (3-14) lnto (3- 11 ) for

subtract1.ng qio as g1.ven by Eq. (3-10) results In the followlng

representation for the induced potent1.al: (X)

rE E;) E3 ;)3 E5 ;)5 ,KI(k) ~) ~.(p,Z) = --(-~ - ~~3 + 2 2 ~ - "')oII(k)Io(kP)cos(kz dk

1 7T 2;)z 2·4 aZ 2·4·6 aZ

Perform1.ng the d1.fferentations results l.n

The series in the brackets is recognlzed as that for I1(Ek). Thus the

potentlal fleld induced by the presence of the tube on a vortex ring

is glven In outer variables by (X)

~ (~p~) - ~ J KI(k) II(Ek) Io(k~p) sin(kz)dk ~i ,z = 7T 0 lI(k) (3-1Sa)

Sectl.on 3-1.3

and in regular varl.ables by

<P. (p,z) 1.

Theory 38

(3-15b)

The total potentl.al for a cl.rcular vortex fl.lament in a tube is, of

course, gl.ven by ip = ip,o + ip l. where (ji 0 l.S given by Eqs. <3-4b) and

(3-6). The solutl.on l.S valid at an instant in time l.n a frame fixed

with respect to the tube.

ST REAM FUNCTION. We defl.ne a stream function '¥ as l.n Eq.

(2-20). Agal.n we wrl.te the stream function as the sum of that for a

circular vortex fl.lament in an unbounded flow, plus the stream

function l.nduced by the presence of the tube:

(3-16)

from the relatl.ons

and Eq. (3- 1 5b) one can deduce the l.nduced stream function:

'l'i(P,z) (3-17)

In a sl.ml.lar manner the stream functl.on for an unbounded vortex ring

can be deduced from Eq. (3-4b):

'l'o(p,z) (3-18)

SectJ.on 3-2. 1 Theory 39

3-2 THE POTENTIAL AND STREAMLINE FIELDS IN THE FIXED FRAME

3-2.1 Inner and Outer Expansions for the Unbounded Ring

In order to compute the constant potential llnes ~nd streaml1nes

for the bounded vortex r1ng we expand ~ 0 (p, z) and '¥ 0 ( p, z) 1nto

two series, one val1d in the 1nner reg10n p < R, and one in the outer

reg10n p > R. we will llkewise expand the lntegral representations

for and ~ (p, z) 1nto ser1es, however only one series 1S required 1

since there are no discont1nuit1es 1n ~. ln the region p < Po 1

NON-DIMENSIONALIZATION. Computations are carried out ln outer

variables for the reg10n p < 1 and z < 1. Consider the following

non-d1mensional forms for the potentlal and stream functlon,

designated with a hat:

(- -) r ~(--) p,z = 4TI ~ p,z (3-19a)

(3-19b)

OUTER EXPANSIONS. We are interested in expansions with which to

compute i 0 and ~ 0 in the region p > E: . Such an expansion is glven A

for ip 0 by Eq. <3-10), resulting 1n the following expression for ~ 0 : --Outer (p > E: ):

1 - ••• ) -:::::r

X

(3-20) Note that as z .... 0, iP 0 .... O. To derive the outer

expanslon for'¥ 0 «(J, 'Z) we wr1te Eq. <3- 11) in outer variables and

expand J1

(kE:) for small E:. Agaln uS1ng the relation given by Eq.

(3-8) we end up w1th an expression slmilar to (3-9), except that the

Laplace transform 1S of J 1(kp) rather than Jo(k~). The resultlng outer

Section 3-2.1 Theory 40

expanSl.on for 'f 0 is:

Outer ( P > € ):

INNER EXPANSIONS. To derive an expansl.on for qi 0 vall.d 1n the

region P ( € we expand Jo (k p) in Eq. <3-6) in the lJ.mit p'" o. Again making use of Eq. (3-8) results l.n:

00 _

-kz Je Jl(kE:)dk o

(3-22)

The integral can be found in tables for Laplace transforms resulting

in the l.nner expansion

Inner (p (E:):

(3-23)

~ + '" r Note that in the l.nner region as z... 0, qio .... -21T and qio'" - '2 as

... l.llustrated l.n Fig. 3-2. The inner expansion for 'fo is derived from

Eq. (3-18) in a sl.ml.lar manner, fl.rst convertl.ng to outer variables,

then expand in! J 1 (kP) for small p. The resulting senes is:

Inner (p (€):

(3-24)

3-2.2 Expansions for the Induced Field

The wall-l.nduced fields can be thought of as resulting from an

"image" vortl.cl.ty distributJ.on outside the tube. The flow within the

tube is therefore potentialand expansl.ons of 'f1

uniformly valid throughout.

and qi are 1

Sectlon 3-2.2 Theory 41

.-Conslder the l.ntegral solutlon for '1' 1. wrl.tten in l.nner

varl.ables. From Eq. (3- 1 7):

(3-25)

We expand 11 (ke), 1, (kt.p' and cos kEZ in the ll.mit of small € The

resul t l.S a series In € whl.ch we wrl.te in the form:

" ~3 (p,z) (3 + ~s (p,z) (s + . (3-26a)

where

1 (b,+ + 2 ° 4 b 2) Y,+ (3-26b)

1 1 (b6 + 2.4 b,+ + 2°42°6 b2) Y6

etc.

The coeffl.cl.ents, bn and Yare gl.ven by: n

-2 -'+ --2 b2 (E-) bl!

p pz = = (zr.t; - 2· 2!)' z '

-6 -'+ -2-'+ P P P Z

b6 = (2204206 - 2204202! + 204!)' •••

00

Yn 2 J KI (k) kn dk

o 11(k)

(3-26c)

(3-26d)

,.. --The same procedure l.S followl.ng in expanding ~i(P'z) We

wrl.te Eq. (3- 1 5) in l.nner variables and expand the integrand for small

e,resultlng In the followl.ng series:

(3-27a)

where " <P3 C2Y2

" 1 (3-27b) <Ps (C,+ + 204 C2) Y,+

" 1 1 <P7 (C6 + 204 C,+ + 2°42°6 C2) Y6 etc.


and

-z

(3-27c)

The integrals Y n defined in (3-26d) must be evaluated numerically.

Because Yn gets large as n increases we compute instead

n-l kn dk = _2_

n! Yn (3-28)

M remains bounded, and has the follow1ng asymptotic form for large n n

[18]: 1 3 1 9 1

Mn = 1T ( 2' + '4 ;: + 16 ;:(n-1) + . • .)

Values of M for n = 2, 4, 6, ... , 100 were numerically evaluated at n

the National Center for Atmospheric Research on their CDC-7600

computer and are glven ln Table 3-1. We might note at this point that before the closed form solut1on

for the potential function was found, we der1ved its expansion using

the method of matched asymptotic expansions with E: as the small

parameter. The inner expanslon is that for the unbounded vortex ring.

The boundary condltion 1S satlsfied in the outer expansion Wh1Ch in

the limlt ~ ~ 0 represents a point dipole at the or1gin. Matching the

two expans10ns at thelr approprlate lim1ts produces the series given

by Eq. <3-27).

3-2.3 The Potential and Streamline Fields

The streamline and potential flelds were computed and plotted ln

the fixed frame. All computations were done using a PDP-11/45*

*The PDP-ll/45 was actually emulating a Danish computer called the RC-4000. I was very kindly given free use of the computer by Dr. John Wilcox and his Solar Physics group in the Institute for Plasma Research, here at Stanford University.

Section 3-2.2

n

2 4 6 8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

M n n

5.0065950 52 2.5142592 54 2.0827486 56 1. 9192741 58 1. 8358962 60 1. 7855132 62 1. 7516317 64 1. 7271765 66 1. 7086378 68 1. 6940734 70 1. 6823163 72 1. 6726200 74 1. 6644831 76 1. 6575556 78 1. 6515855 80 1. 6463865 82 1. 6418178 84 1. 6377712 86 1. 6341616 88 1. 6309220 90 1. 6279980 92 1. 6253456 94 1. 6229285 96 1.6207169 98 1. 6186854 100

TABLE 3-1

Numerically computed values for M • n

Theory 43

M n

1.6168130 1. 6150816 1. 6134759 1.6119827 1.6105904 1. 6092893 1. 6080706 1. 6069268 1.6058511 1. 6048377 1. 6038812 1. 6029771 L 6021210 1.6013094 1. 6005387 1. 5998061 1. 5991086 1. 5984440 1. 5978098 1.5972041 1.5966250 1. 5960707 1. 5955397 1. 5950306 1. 5945420


computer and an HP-7221A plotter. Developing algol procedures to

calculate the cylindrical harmonics, Eqs. (3-20) and (3-21) were used .....

to compute the potential and streamlines in the outer region (P) € ), .....

and Eqs. <3-23) and <3-24) in the inner region ( P)€ ) of the

unbounded circular vortex f1lament. "" ""

To compute the induced f1eld, the expanded forms of '!' i and ~ i

as given by Eqs. (3-26) and (3-27) were used along with the

numerically calculated values for M given in Table 3-1. The total n

potential and streamline fields are the sums of the unbounded and

induced fields Wh1Ch were calculated separately. All computat1ons

were performed in outer variables over a 51x51 point grid in the

region 0 $ (P&~) :s: 1, and extended to the region -1 ~ Z ~ 0 by symmetry.

Plots were made in the region 0 ::: p~ 1, _1 ::z :: 1; half of the tube is

shown in the figures.

The streamlines for an unbounded vortex r1ng in non-steady

coord1nates were prev10usly sketched 1n Fig. 2-5. For comparison with

f1gures to follow we show in F1g. 3-3 the streamlines (solid lines)

and constant potent1al lines (dashed) for an unbounded vortex r1ng of

Slze € = R/po= 0.4. As demonstrated by Eq. (3-10), 1n a fixed frame

of reference the non-steady outer f1eld is dipolar in character with

higher order harmonics becoming less and less important as P + 00

( 1. e ., as E: +0).

Compare this wlth the potential and streamline fields for the

same size vortex ring, but now bounded by the tube walls as shown in

F1g. 3-4. Although the flowfield still has a d1pole character, the

flow 1S forced to turn parallel to the tube wall. The effect is a

generally more aX1al flow. As the core is approached the streamlines

approach those of the unbounded ring shown in Fig. 3-3. The constant

potential lines must be perpendlcular to both the boundary and the z

axis for a bounded ring, resultlng in two streamlines which extend to

inf1nity in Oppos1te directions. A

The induced f1elds, the SOlld llnes represent1ng constant ~ A 1

and the dashed 11nes constant ¢., are shown in Fig. 3-5, again for 1

Section 3-2.3 Theory 45 z

t

----

FIGURE 3-3

The streamlines and constant potential lines in the fixed frame

for an unbounded vortex ring with E = 0.40.

Section 3-2.3 Theory 46 z

FIGURE 3-4

The streamlines and constant potential lines in the fixed frame for a bounded vortex ring with E = 0.40.

Section 3-2.3 Theory

z

t ---..- ...

--------------------.-----

------ ..... --..

------------------

----.----- -------- .... - -- ~--

--------- .... ----

----------------_ ... --------------

--------------------_ .... -

... ------.-

-----

FIGURE 3-5

The wall-induced streamline and constant potential lines in the fixed frame for a bounded vortex ring with £ = 0.40.

47


E=O.40. This field added to the unbounded field of Fig. 3-3 produces

the pattern which was g~ven in Fig. 3-4. Note that the induced

veloc~ty field acts in the oppos~te direction to the direction in

which the vortex ring moves. Thus the tube acts to slow the vortex

ring as compared with the same ring in an unbounded flow.

3-3 THE VELOCITY OF THE VORTEX RING IN A TUBE

As shown by the induced streamlines in Fig. 3-5 the velocity of

a vortex ring in a tube is that which it would have were it in an

unbounded flow, less the velocity induced by the presence of the tube:

U = U 0 - Ui <3-29 )

where Uo is the velocity of the unbounded vortex ring and U1 the

wall-induced velocity given by:

U. = I _! a~i(p'Z)J -~ R dZ Z = 0, p = 1

Applying the derivative to Eq. (3-27) and using the numerically

calculated values for M results in the following power series in E2: n

U. = 4;R (5.006595 E 3 + 2.828542 E 5 + 2.440721 E 7 + ••• ) ~

(3-30)

Note that U. depends only on r and not on the details of the ~

vortic~ty distribut~on in the core. This is because the induced

velocity is ~n the far field of the ~mage vortex system residing

outside of the tube. We non-dimensionalize the velocity as follows:

"-and wr~te U ~n the form: ~

0-31)

Sectl.on 3-3 Theory 49

<3-32)

10 1 The calculated values for the coeffic~ents up to O(c ) are given in

Table 3-2.

RADIUS OF CONVERGENCE. Consider the power series P = ~a cn n

By the d'Alembert ratio test the radius of convergence Co ~s g~ven by

Co lim I an_ 1 I n + 00 a (3-33)

n

Examining the coeff~c~ents ~n Table 3-2 for large n suggests a rad~us

of convergence of 1. We show this graphically w~th a Domb-Sykes plot

[71]. Assuming that near a singularity the power series can be

represented as

a. (Eo -E) (3-34)

we expand using the b~nomial theorem in powers of 1/n to produce:

(3-35)

As n gets large this ratio approaches 1/Eo linearly with 1/n. At 1/n

approaching zero, therefore, the intercept yields the radius of

convergence, while the slope ind~cates the type of s~ngularity.

The Domb-Sykes plot for U. is g~ven in Fig. 3-6. There is ~

little doubt that the resulting curve approaches the ordinate at

a 1/a = 1 with a slope of 1. We conclude, therefore, that the power n+ n

series for U. converges for values of E up to 1 (i.e., for any size ~

vortex ring), with a simple pole at E = The decrease in r~ng speed due to the presence of the tube is of

Small vortex rings would not be much affected while

larger rings might be substantially affected by the boundaries. In

Fig. 3-7 U./Uo is plotted as a function of E for different values of " ~ Uo. We express U 0 in terms of aiR as def~ned by the Kelvin/Lamb

Section 3-3

'" u n n

3 5.006595 5 2.828542 7 2.440721 9 2.296007

11 2.223716 13 2.180712 15 2.152066 17 2.131493 19 2.115936 21 2.103728 23 2.093875 25 2.085750 27 2.078929 29 2.073121 31 2.068113 33 2.063750 35 2.059915 37 2.056517 39 2.053484 41 2.050762 43 2.048303 45 2.046073 47 2.044039 49 2.042178 51 2.040469

'" U n n

53 2.038892 55 2.037434 57 2.036082 59 2.034824 61 2.033650 63 2.032554 65 2.031526 67 2.030562 69 2.029655 71 2.028800 73 2.027993 75 2.027230 77 2.026508 79 2.025823 81 2.025172 83 2.024554 85 2.023965 87 2.023404 89 2.022869 91 2.022357 93 2.021868 95 2.021400 97 2.020951 99 2.020521

101 2.020108

TABLE 3-2

Coefficients for the wall-induced velocity of a vortex ring in a tube.

Theory 50

Section 3-3 Theory 51

1.0

0.5 a 1.0

1 n

FIGURE 3-6

A

The Domb-Sykes plot for U .• 1.


1.0 ~ =.35 .15 R

.000001

0.5 1.0

E

FIGURE 3-7

The wall-induced velocity of a vortex ring in a tube.

Uo is given by the Kelvin/Lamb formula.

Section 3-3

Z i r 0 ~ ~ • +

1 V

+

t

1 FIGURE 3-8

The analog of a vortex pair in a channel.

i) only the first images are shown.

ii) the upper arrows indicate the induced velocity from the opposite vortex, while the lower arrow is that for the image vortex.

iii) the large arrows indicate the direction of motion of the vortex pair.

t G

1

Theory 53


formula for the velocity of a thin core vortex ring with constant

vorticity in the cor€ [see Sect~on 2-3.1, Eq. (2-15)]:

_ r 8R 1 U 0 - -(in - - -)

4rrR a 4

Note from F~g. 3-7 that for a given Uo (or aiR), as the vortex ring

increases ~n size, the wall induced velocity increases until it

reaches a value equal to Uo• At this point the vortex ring does not

move relative to the tube, and larger rings actually move backwards I

This can be understood by considering the two-dimensional analog to

the vortex ring in a tube: a vortex pail" in a channel. As shown in

Fig. 3-8 as the distance between the vortex pail" increases (i.e., as

8 increases) the image vortices move closer to the wall, until a

point is reached at which the velocity induced on one line vortex by

its image is the same as that induced by the other line vortex. As ,8

increases further the image system induces an even greater velocity on

the vortex pail" causing it to move backwards. We note also from Fig.

3-7 that in comparing two vortex rings of the same size the faster

moving, and therefore more energet~c vortex ring is affected less by

the presence of the tube.

3-4 THE POTENTIAL AND STREAMLINE FIELDS IN THE MOVING FRAME

3-4.1 The Kinematic Relationships Among Vortex Ring Parameters

Consider for a moment the dipole representation for a sphere in

a uniform flow. It ~s well known that a point dipole of strength r embedded in a uniform flow with velocity U has a streamline pattern

which represents an inv~scid, potential flow around a sphere which is

moving with constant velocity through a stagnant fluid. The size of

the sphere can be adjusted by varying the dipole strength r or the

free stream velocity U. For a given U the sphere radius R' increases

Sec hon 3-4. 1 Theory 55

as r increases, however for a given dipole strength, the sphere S1ze

decreases as U increases. In other words, there is a kinematic

relat10nship among H', r, and U.

Analagous to, but more comp11cated than the d1pole in a un1form

flow 1S the flowf1eld produced by a vortex r1ng. In a frame f1xed 1n

space the flow has a bas1cally d2pole character, w1th h1gher order

poles accounting for the f1nite size of the vortex r1ng. However,

when one takes into account the vortex r1ng velocity by transferring

to a coord1nate system movlng with the rlng, a spher01dal volume of

flu1d accompany1ng the vortex core 1S ident1f1ed. Just as w1th the

mov1ng point d1pole, the extent of this spheroidal volume is related

kinematically to the strength r and speed U of the vortex r1ng, and

as wi th the d1pole, an increase 1n r or a decrease in U resul ts in an

increase in the volume of fluid within the spheroid. Unllke the point

dlpole, however, the vortex rlng has a f1nite radius H, resulting in a

spheroidal rather than a spherical reglon of flu1d w1th1n the vortex

rlng.* We m1ght also expect that the f1n1te extent of the core, and

the vortic1ty distr1bution w1thin the core would have an effect on the

volume of the vortex ring. However, as shown in the next section,

th1s effect 1S very slight.

For a vortex ring 1n a tube a purely k1nematlc relationsh1p also

eX1sts among the vortex rlng parameters, but w1th add1t10nal effects

due to the boundaries. In the last sectlon we analysed the effect of

the tube on theveioc1ty of the vortex r1ng. We will assess here

changes in vortex ring size and shape result1ng from the presence of

the boundary. In add2tion, we compute and plot the kinematic

relationship among non-d1mensional parameters describing the vortex

ring strength, velocity, size, and shape so that flow visualization

measurements can be used to compute the total circulation of typical

*We refer to the whole spheroidal volume as the "vortex ring" and the toroidal region in which vorticity is concentrated as the vortex core.


vortex rings in a tube or in an unbounded flow, as a function of

time.-

Let cP and If' be the potential and stream functions in a frame

fixed to the tube. These functions can be further divided ~nto

unbounded and induced fields as given by Eqs. (3-2) and (3-16), and

non-dimens~onal~zed as in Eq. (3-19). The potential and stream

functions ~n a frame moving with the vortex ring we give the subscript

s, and can likewise be divided into the unbounded and induced fields.

They are given by:

In

cP s

If' s

non-dimensionalized

A

cP s

A

If' s

= cP - uz (3-36a)

(3-36b)

form they become:

A 1 A

= cP --u z 8

(3-37a)

A 1 A -2 = If' +-2 u p

28 (3-37b)

where U is the non-dimensionalized ring velocity as given by Eq.

<3-31) .

CHARACTERIZATION OF RING VELOCITY. Ultimately the velocity of

the vortex ring must be specified either by a calculation which takes

the dynamics of the flow (~ncluding the vorticity distribut~on) into

consideration or, as we propose to do here, by an experimental

measurement. For the purposes of determining ~ts relat~onships with

other parameters, however, we must vary U in a convenient way. We

*For a discussion of the motivation and concepts behind this calculation please read Section 2-5.


begin by d1v1ding it into an unbounded and 1nduced velocity as given

by Eq. (3-31):

A

U = U - U. o 1

i) U is determined from Eq. (3-32) and depends only on the Slze of 1

the vortex r1ng relat1ve to the tube.

11) 60 1S expressed more conveniently using the core parameter aiR as

defined by the Kelv1n/Lamb formula extended to a continuous

vort1c1ty d1stribution [see Sect10n 2-3.1 and Eq. (2-17)]:

A 8R 1 Uo = ~n - -

a 3

W1th this expression we use aiR to specify Uo •

(3-38)

We should be careful to note that for the purposes of

determining the kinemat1c relationships in which we are 1nterested,

Eq. (3-38) 1S only a conven1ent defln1tion, and aiR 1S used only as a

parameter to character1ze U o ' Of course, to the extent that the

Kelv1n/Lamb formula is accurate (see Section 5-2.2) aiR does have

physical slgnificance as a measure of an effective core radius to the

radius of the vortex ring, but its use here should not necessar1ly

1mply its acceptance as an accurate representation of the velocity of

a real vortex r1ng.

POTENTIAL AND STREAMLINES. In Figs. 3-3 and 3-4 we compared the

non-steady streaml1nes for an unbounded and bounded vortex r1ng w1th

£ = 0.40. Streamllnes and constant potent1al lines in a frame of

reference attached to the vortex ring are shown in Fig. 3-9 for a A

vortex r1ng with £ = 0.40 and aiR = 0.20 (U o = 3.356). F1g.3-9a

dep1cts the vortex ring in an unbounded flow, and 3-9b the same ring*

in a tube. Streamlines are shown as solid llnes and constant

*By the "same ring" we mean a vortex ring with the same radius (i.e., £ or R), the same strength, r, and moving with the same unbounded velocity, Uo (i.e., having the same value of aiR).


FIGURE 3-9

The streamlines and constant potential lines in the moving frame for a bounded and unbounded vortex ring with £ = 0.40.

(a) unbounded vortex ring aIR = 0.20


FIGURE 3-9

The streamlines and constant potential lines in the moving frame for a bounded and unbounded vortex ring with E = 0.40.

(b) bounded vortex ring aiR = 0.20


potential lines are shown dashed. For reference to the Kelv~n/Lamb

formula, the streamline correspondng to the designated value of aiR is

darkened.

We note f~rst the three reg~ons commonly observed when a vortex

ring is visual~zed with dye.

1. The core region, which in these calculations is concentrated along

a circle. If you use the KelvinlLamb formula as an indication of

core size the core would be roughly within the darker streamline

correspond~ng to aiR = 0.20.

2. The spheroidal volume of fluid which accompanies the core. The

extent of this region ~s delineated by the outermost closed

streamline.

3. The outer ambient fluid through which the vortex ring propagates.

The streamlines in this region extend to infinity.

Second we note that the streaml1nes for the bounded vortex ring become

parallel and the constant potent~al lines perpendicular to the tube

wall as p ~1, whereas this is not the case for the unbounded vortex

ring. As to the size and shape of the vortex ring, the bounded vortex

ring is slightly thicker than the unbounded ring, giving it a sl~ghtly

greater volume, although for a vortex ring of th~s size the difference

1S not great.

In order to see more dramat1cally the effect of the tube on the

size and shape of the vortex ring cons1der the extreme case of a very

large vortex ring. In F~gs. 3-10 are shown the streamlines for

bounded and unbounded vortex r1ngs w1th E = 0.70 and aiR = 0.25. Due

to viscous effects at the tube wall, such a large vortex r~ng

(relative to the tube) is not physically realistic, but it shows

dramatically the change in S1ze resulting from the presence of

boundaries. The thickness of the vortex ring is increased, result1ng

1n a larger volume of fluid carr1ed along w1th the vortex core;

however, this volume of fluid moves at a slower speed. The outer

radius R' of this extremely large vortex ring is slightly less as a

result of the tube.


FIGURE 3-10

The streamlines and constant potential lines in the moving frame for a bounded and unbounded vortex ring with £ = 0.70.

(a) unbounded vortex ring


, I II II

FIGURE 3-10

The streamlines and constant potential lines in the moving frame for a bounded and unbounded vortex ring with E = 0.70.

(b) bounded vortex ring

Sect~on 3-4.1 Theory 63

THE KINEMATIC RELATIONSHIPS. A more detailed description of the

effects of the tube on the shape of the vortex ring w~ll result from

calculations of the kinematic relationships among the parameters wh~ch

characterize the s~ze, shape, strength and veloc~ty of the vortex

ring. These parameters, ~n non-d~mens~onal form are:

i) "- U U = =-:~-r/41TR

ii) E: = R/po

"- "-

Uo- U. l. where

iii) T/R

"-

U. l. U. (E:) and U 0 l. U 0 (a/R)

iv) R' /R

Aga~n we po~nt out that aiR ~s used here only as a parameter to

character1ze U so that an unbounded vortex r1ng which is o , "-

subsequently placed 1n a tube will have the same value of UO, and

therefore the same value of aiR. "-

For a vortex r~ng with g~ven Uo and E: (and therefore

have calculated the stream funct~on ~ and potential funct~on s

Eqs. (3-37a and b). The th1ckness TIR was then determ1ned "-

evaluated at p =0, and R' IR from '1' evaluated at z=O·. s

U.), we "-~

 from s "-

from s

In Fig. 3- 1 1 is shown the variat10n of TIR w1th Uo (01" aiR) for "-

E: = 0, 0. 1, 0.2, ... , 0.7. Note that Uo decreases wlth increasing

aiR. The SOlld curve corresponds to E: = 0, the unbounded vortex ring,

and the dashed curves to 1ncreas~ng values of E:. We observe an "-

upward trend in TIR as aiR increases, that ~s as Uo decreases.

Analagous to the dipole 1n a uniform flow (as discussed at the

beginning of this section), this shows that for a vortex r~ng of given

rad~us, the th~ckness (and volume) increases w1th e1ther decreas~ng

ring veloc~ty 01" ~ncreasing r~ng strength. Related to th1S, we note

also that the tube has a greater ~nfluence on vortex rings with "-

smaller values of Uo That lS, deviat10ns from the soll.d curve are

*T d h . . h . dR' d correspon s to t e max1mum 1n s along t e z aX1S an correspon s to the zero along the p axis (away from p = 0).

Section 3-4.1

1.10

T R

0.90

0.70

0.50

I I

I I

I

,. ,. ,. ,.

0.10

,. ,.

FIGURE 3-11

,. ,.

,. ,. ,. ,. ,. ,. ,. ,. ,.

,. ,. ,. ,.

, ,. ,.

0.20

A

0.30 aIR

The variation of T/R with Uo for different values of E.

Solid curve: unbounded vortex

Dashed curves:

1st: E = 0.30

2nd: E = 0.40

3rd: E = 0.50

4th: E 0.60

5th: E 0.70

Theory 64

0.40

R' R

Section 3-4.1

1.45

1.40

1.35

1.30

1. 25 0.1 0.2 0.3

aiR

FIGURE 3-12

A

The variation of R'/R with Uo for different values of E.

Solid curve: unbounded vortex ring

Dashed curves

1st: E = 0.50

2nd: E 0.60

3rd: E = 0.70

Theory 65

0.4


grea ter for slower and for stronger vortex rings. For £ S. 0.3 there

is little influence of the boundary on the thickness of the vortex

ring. As £ increases, however, the effect of the tube becomes more

and more apparent as an increase in the thickness and volume of the

vortex ring over the same r1ng in an unbounded flow. (',

The var1ation of R'/R with Uo as a function ofE is shown in

Fig. 3- 1 2 where again the solid curve corresponds to the unbounded

vortex ring and the dashed curves to increasing values of E. For

values of E up to about 0.4 the effect of the tube on R' IR is

negligible and for values up to 0.50 slight. It is interesting that

the effect of the tube is to 1ncrease R' for large values of U but '" 0

decrease R' for small values of Uo' As E increases, the point at

wh1ch the dashed curves cross the curve for an unbounded ring moves to

the left. That is, for very large vortex rings, as in Fig. 3-10, the

effect of the wall is to decrease R' over an identical, but unbounded

vortex ring.

We p01nt out once again that the relationships plotted in Figs.

3-11 and 3-12 are purely kinematic. Thus, measurements can be used 1n

conjunction with these results to calculate quantities of 1nterest.

In Chapter 5 we discuss the use of flow visua11zation measurements

together with these kinemat1c relationships to calculate the total

circulation of vortex rings.

3-4.2 The Effect of a Finite Core on Ring Shape

As was discussed in detail 1n Section 2-5 we have made the

assumption that in calculating the relationships between the shape and

other parameters of the vortex ring we can neglect the fin1te extent

of the core and concentrate the strength of the vortex ring along a

circle. In th1s sect10n we perform a calculat10n to test this

hypothesis.

Sect~on 3-4.2 Theory 67

A vortex ring with E = 0.40 and aiR = 0.25 was chosen for the

test.* Thirty-seven vortex filaments were distr~buted ~n a roughly

c~rcular area with radius b such that blR = 0.30. The vortex

f~laments were weighted according to the vort~c~ty d~str~bution ~n a

real vortex ring as measured by Maxworthy [36, F~g. 6]. The sum of

strengths of the individual filaments is the total c~rculat~on of the

vortex ring. Using the l~near~ty of the B~ot-Savart law (see Section

2-1) the potent~al f~eld was computed by add~ng the fields of the

~ndividual vortex filaments.

The constant potent~al lines for a bounded vortex r~ng with the

total c~rculation concentrated along a single vortex f~lament are

compared with the vortex ring having a finite core in F~g. 3-13. The

37 vortex filaments over wh~ch the circulation of the vortex ring ~s

distributd are indicated with dots.

A careful comparison of the constant potent~al l~nes (e.g.,

using a l~ght table) shows that d~fferenes ~n vortex ring shape

between the f~nite core vortex r~ng and circular vortex f~lament are

extremely small. There appears to be a very sl~ght bulge ~n the

vortex ring with the f~nite core, but T and R' are v~rtually

unchanged. Even near the core where one m~ght expect greater

d~fferences they are slight. Th~s would suggest that for the purposes

of comput~ng the streaml~ne and potential f~elds outs~de of the core

of a vortex ring, a vortex f~lament model ~s quite accurate. In

computations of the velocity of a vortex, of course, the size of the

core rema~ns a cr~t~cal factor (Section 2-3). A compar~son was also

made between unbounded vortex rings, but with the same result as the

bounded case.

*These values were chosen because most of our experimentally produced vortex rings had values of E below 0.4 and calculations based on the Kelvin/Lamb formula suggest that for these rings, aIR ~ 0.25 (Section 5-3).


FIGURE 3-13

Constant potential lines for a vortex ring with a finite core compared with a circular vortex filament.

(a) finite core

68


FIGURE 3-13

Constant potential lines for a vortex ring with a finite core compared with a circular vortex filament.

(b) circular vortex filament

E9

4 Experiment

lUg wlWt.t6 ha.ve Ut:te.e wlWt.t6 wfUc.h 6eed on :thcWL velowy; .u.:t:tle wh.vt-t6 have .6ma.U:eJL wfuJt.t.6 a.nd .60 on :to vv.,C.O.6Uy.

- L. f. Ric.hoJr.d.6 0 n

Complementing the theoretical calculations, an experiment was

performed to observe real vortex rings as they propagate up a tube.

We comb1ne qualitative observations w1th flow visual1zation

measurements to study the characteristics and development of vortex

rings from very low to very high Reynolds numbers. In Chapter 5 these

measurements are comb1ned w1th the kinematic relat10nships described

in the last chapter to calculate the total circulation of our

exper1mentally produced vortex rings as a function of time.

4_1 APPARATUS

BASIC APPARATUS. The bas1c apparatus is shown schemat1cally 1n

Fig. 4_1. A plexiglass tube w1th an internal diameter of 11.88 cm., a

wall thickness of 1 cm., and a total height of 55 cm. 1S secured

vertically on a heavy, cast iron mount. Seven circular orif1ces of

different size, 2.5 cm. thick and tapered at an angle of 60 0 are

mounted 1n the tube. Dp-f1ning d as the orifice diameter and D as the

tube diameter, the orifice S1zes used in the experiment are:

diD = 0.10, 0.15, 0.25, 0.32, 0.42, 0.48, 0.64.

By mounting two Endevco accelerometers on either side of the tube it

was discovered that when filled with water the tube exhibited a

70

Section 4-1

2.5 em

FIGURE 4-1

40 em

x p

11. 88 em

D

d~

Basic experimental apparatus.

Experimen t 71

Travel dial

~ Potentiometer

Section 4_1 Experiment 72

cantilever mode of vibration with a natural frequency of roughly 35

Hz. To dampen out these unwanted oscillations a heavy metal support

was added to the top of the tube.

A single pulse of water is ejected through an orifice by an

hydraulically activated p1ston. The p1ston speed and stroke length

are adjustable. A linear potent10meter is attached to the piston to

monitor its x -t characteristics, and a travel dial is used to measure p the stroke length X. From these measurements the average piston

p speed u is calculated, and from

p u

p (d/D)2

(4-1)

the slug flow, or "jet" velocity determined. Defining the jet

Reynolds number as

U d Re. = ~

J v (4-2 )

and comb1ning w1th Eq. (4-1) we see that there is a general 1ncrease

in Re. with decreasing or1f1ce S1ze. J

EXPERIMENT. The piston speed is adjusted to produce tube

Reynolds numbers of roughly 5000, 4000, and 3000 where

Du Re = --P..

T v (4-3 )

The piston travel time 1S held roughly constant at about 0. 1 70 sec;

thus more fluid 1S ejected at the higher Reynolds numbers.

In F1g. 4-2 the ma1n components of the experiment are sketched.

The potentiometer output, a d1rect indication of x (t), is displayed p

on a Tektronix oscilloscope and recorded photographically. Either an

accelerometer attached to the piston or a pressure transducer mounted

flush w1th the tube wall 1S used to tr1p the oscilloscope. It is

found that, except at the lowest values of ReT' the xp-t slope is

dyed water

FIGURE 4-2

-.. :;::"~'

~r@w~~

clear water

pressure transducer

accelerometer

potentiometer

The main components of the experiment.

Hydrogen Bubble Generator

----

- l50v

t-. ta anades

Ul ro n rt f-'. o ::l .pI ....

t>l ~ '0 ro 11 f-'. S ro ::l rt

-...J W

Section 4_1 Experiment 74

nearly l1near, indicat1ng constant p1ston speed. An example is shown

in Fig. 4-3a. The upper curve is output from an accelerometer

attached to the piston show1ng piston acceleration and deceleration

periods of about 15 msec. For the lowest values of ReT' the xp-t

character1st1cs are as shown 1n Fig. 4-3b. We approximate this piston

trajectory as 1/4 of a sine wave.

In order to visualize the flow, either dye is inserted into the

chamber below the orif1ce or hydrogen bubbles are produced from thin

rings, 1/16 inch in diameter, which just fit on the underside of the

or1f1ce (see F1g. 4-2). The r1ngs are insulated and wrapped in a

spiral fashion w1th 2 m1l plat1num W1re to serve as the cathode 1n the

electrolysis process. Two th1n strips of 5 mil plat1num sheet are

glued vertically to oppos1te sides of the tube for the anodes. About

25 cc. of sodium sulfate was d1ssolved in the water as an electrolyte

and 150 volts applied between the anode and cathode. Tiny hydrogen

bubbles are swept into the vortex ring as 1t forms, all

red1ssolv1ng except for those trapped 1n the center of the core, a

relat1vely stagnant region. In th1s way bubbles can be used to

measure R, the radius of the vortex ring to the core center11ne.

The use of dye, of course, allows v1sualization of the total

extent of the vortex ring. Hed food coloring was injected into the

chamber below the or1fice to a concentration of about 2% by volume.

Dye gives the best descr1ption of the format10n process as well as

qua11tative features such as the heav1ness of the wake behind the

vortex ring. Quant1tative measurements of the shape of the vortex

r1ng (i.e., T and H') are made uSIng the dye visualization.

Unfortunately, although the core region 1S usually darker than the

outer region of the vortex ring, 1t usually does not contain enough

detail to determ1ne with prec1sion the locat10n of the core

centerline.

MOVIES. H1gh speed 16 mm. color mOV1es were taken for all

comb1nat1ons of or1f1ce size and tube Reynolds number. Over 190

Section 4-1

FIGURE 4-3

Piston trajectory characteristics. 20 msec/div.

Experiment 75

(b)

(a)

Section 4-1 Experiment 76

sequences, or "runs," were filmed of vortex rings ranging in Reynolds

number from about 690 to 50100. These mov~es were subsequently

analysed to obtain both qualitative and quant~tative data.

With the hydrogen bubble techn~que l~ghting ~s a major problem.

The intens~ty of the reflected l~ght from the tiny bubbles ~s very

sens~tive to the l~ghting angle, and the round tube added unwanted

reflections. By adJust~ng the l~ghts so as to come from above the o movie camera at an angle of roughly 10 from the vert~cal, we were

able to obtain hydrogen bubble movies for all but the slowest vortex

rings. When using the dye the lighting was from behind the tube and

reflections were not a problem.

In order to accurately determ~ne the film speed, a four stage

timer was constructed i (see F~g. 4-2) consisting of four divide-by-ten

integrated circuits connected to light-emitting d~odes so as to be

visible in the movies. The diodes, wh~ch have a rise time of about 5

microseconds, are arranged ~n four c~rcles of ten each, each c~rcle

representing a dec~mal place. A sine wave generator provides the

power and cal~brat~on for the timer. W~th the t~mer v~sible in the

movies, very accurate cal~bration of film speed was obtained. For

most runs the f~lm speed was about 100 frames per second.

4-2 QUALITATIVE OBSERVATIONS

As dyed water ~s ejected through the orif~ce we observe the

rollup of the emerg~ng cyl~ndr~cal vortex sheet ~nto a vortex r~ng

(see Sect~on 2-4), ~ts detachment from the orif~ce, and ~ts

propagat~on up the tube. In some cases the vortex ring can be

*By William Janeway to whom we are indebted for a great deal of help with the electronics and experiment.

Sec hon 4-2 Experl.ment 77

observed to veer from the tube axis and l.nteract with the tube wall as

well.

SMALLEST ORIFICES. The smallest orifices (diD = 0. 1 0, 0.15)

produce the smallest and most energetl.c vortex rl.ngs. Assocl.ated wl.th

the hl.gh jet Reynolds numbers, the vortex rl.ngs have very high values

of rl.ng Reynolds number and are turbulent l.n character. For these

small orl.fl.ces the formation of a vortex rl.ng l.S followed by the

production of a turbulent Jet.* The vortex ring l.nl.tl.ally occupies

the tl.P of the Jet, but due to the concentrated nature of the

vortl.cl.ty wl.thl.n the ring, soon breaks away and travels l.ndependently

up the tube. After the pl.ston has stopped, the Jet qUl.ckly diffuses

and spreads. The vortex ring, on the other hand, travels rapl.dly up

the tube followed by a fal.rly heavy, turbulent wake.

A photograph and drawl.ng of turbulent vortex rl.ngs are shown l.n

Fl.gs. 4-4 and 2-7b respectively. Wl.thl.n the vortex ring we observe a

somewhat discernl.ble core regl.on surrounded by a ll.ghter regl.on of

turbulent flul.d. Wl.th the hydrogen bubble visuall.zatl.on, short

wavelength falrly small ampll.tude waves can be observed along the

core. In addition, a bendl.ng mode of oscillatl.on is sometl.mes

observed; as the vortex rl.ng travels up the tube, the ring of bubbles

defl.ning l.ts core l.S observed to bend back and forth symmetrically

about the ring axl.s.

INTERMEDIATE ORIFICES. The intermedl.ate sl.zed orl.fl.ces (diD = 0.25, 0.32, 0.42) are assocl.ated wl.th l.ntermedl.ate values of Jet

Reynolds number, l.ntermediate rl.ng Reynolds numbers, and

intermediate size vortex rl.ngs. For an example of an "l.ntermediate"

vortex ring, see the photograph at the beginning of this report (just

*During the piston motion, a jet of dyed water is indeed observed, but after the piston has stopped this turbulent region continues to move and develop. In what follows we call this turbulent region a "jet," although it may not at all times correspond to an actual jet.

Experiment 78

FIGURE 4-4 Example of a vortex ring at the highest Reynolds numbers. Note the formation of a turbulent jet behind the vortex ring.

Sectlon 4-2 Experlment 79

before the abstract). As can be observed from thlS photograph, these

vortex rings tend to be well formed, lamlnar, and ln general appear to

be quite stable. The core reglon lS typlcally well deflned, and the

vortex rings propagate wlth 11ttle or no observable wake.

LARGEST ORIFICES. The largest orlflces (diD = 0.48, 0.64)

produce the slowest vortex rlngs wlth the lowest Reynolds numbers. A

result of V1SCOUS dlffusion and convectlon of vortlclty, dye lS

observed ln a lamlnar wake reglon behind the vortex rlng (see Flg.

2-7a). An lnterestlng phenomenon lS observed wlth the vortex rlng

fllmed at the lowest Reynolds number. Apparently a result of

InteractIon WIth the free surface In combinatIon wIth the tube wall,

the extremely slow, clearly viscous vortex rIng IS observed to bend

back as it approaches the top of the tube. That IS, the symmetry

plane (the z=O plane of Flg. 2-3) takes the shape of a spherlcal cap,

nose foreward. As the vortex rlng encounters the free surface the

cap flattens out agaIn, and the rlng breaks down.

In general, when the vortex rings encounter the free surface the

radlus qUIckly lncreases, followed by a rapId breakdown.* the vlsual

result, especlally for those vortex rlngs wlth llttle wake, lS a dyed

region of fluld at the bottom of the tube where the rlng formed, a

dyed region at the top of the tube where the ring collapsed, and clear

water ln between. The smallest vortex rIngs Wlth very hlgh Reynolds

numbers, however, had suffIcient energy to propel them completely out

of the tube.

TRANSITIONAL VORTEX RINGS. Between the three groups of vortex

rings which were labeled above as coming from the smallest,

intermedlate, and largest orifIces, are "transltl0nal" vortex rlngs.

*The increase in radius is understood by considering the image vortex ring on the opposite side of the free surface. For beautiful photographs of the breakdown of a vortex ring at a surface, see Magarvey and MacLatchy (33).

Section 4-2 Expenment 80

These rlngs are characterlzed by two major features. First, one

observes in these vortex rlngs characteristlcs from both the hlgher

Reynolds number and lower Reynolds number rings they separate.

Secondly, these translt10nal vortex r1ngs generally show more signs of

large amplltude instab1l1ty than do the others. For example, the

trans1tional vortex r1ngs between the hlgh Reynolds number/small

orifice and the intermedlate Reynolds number/intermediate or1fice

vortex r1ngs (d/D ~ 0.15, 0.25) show slgns of generally larger scale

turbulent mot1ons, but v1sually the development of turbulence appears

to be somewhat intermittent. Relat1ve to the rings at the highest

Reynolds numbers, they generally exh1b1t larger ampl1tude osclllations

as well. One such transitional r1ng exhiblted a "rocking" type of

lnstabllity (see Section 4-4.3), and another a large amplitude bendlng

mode of oscillation. The transitional vortex rlngs between the

intermed1ate and the largest size oriflces (d/D ~ 0.48), on the other

hand, show slgns of large amplitude lnstability in the sense that a

greater number of these rlngs would rather abruptly move towards, and

lnteract wlth the tube wall. At these Reynolds numbers other

researchers have observed the development and growth of large

amplltude fairly long wavelength waves along the core centerline

(Sectl0n 2-4.2).

INTERACTION WITH THE TUBE WALL. The interactlon of a vortex

rlng w1th the tube wall 1S an lnterest1ng phenomenon. Except for the

vortex r1ngs produced wlth an orlf1ce Wh1Ch was intentionally made

very irregular (Section 4-4.3), the location where the r1ng collides

with the wall appears to be random. As a vortex ring makes contact,

one side tends to remaln flxed, or "stick" to the wall whlle the other

rotates around.* From the hydrogen bubble visualizatlons one observes

*Again this can be understood by visualizing the image system with the 2-D analog of Fig. 3-8.

SectJ.on 4-2 Experiment 81

that when the core of the vortex rJ.ng touches the wall, vortex

stretchJ.ng manJ.fests itself as a rapJ.d tWJ.stJ.ng of the vortex core

before the ring collapses into a generally turbulent mass of water.

Subsequently, thJ.s turbulent, vortJ.cal region J.S observed to rotate

away from the tube wall, apparently the result of the cancellatJ.on of

some vorticJ.ty by vJ.scous forces at the wall, resultJ.ng J.n a net

vortJ.cal motJ.on away from the wall.

4-3 QUANTITATIVE RESULTS

4-3.1 Data Acquisition

Data was reduced for 26 sequences or "runs" uSJ.ng both the dye

and hydrogen bubble vJ.sualJ.zatJ.on techniques. The runs are numbered

from 1 to 26 J.n the sequence J.ndJ.cated J.n Table 4_ 1 • The values for

ReT gJ.ven J.n thJ.s table are rough and should be used only as a gUJ.de.

The calculated values of ReT' more accurate values of diD, Rej

, and

other J.nformatJ.on assocJ.ated wJ.th each run are given 1n Table A-1 of

AppendJ.x A. The fJ.rst step in the acquJ.sJ.tJ.on of data was to project

each run onto a large sheet of graph paper and, by advancing the

projector frame at a tJ.me, carefully outlJ.nJ.ng the vortex ring with as

much detail as possible at many locatJ.ons J.n ltS evolutJ.on. WJ.th the

4-stage timer, the fJ.lm speed was accurately determJ.ned for each run.

Thus, knowJ.ng the number of frames between each outlJ.ne, we were able

to determJ.ne the tJ.me history of the vortex rJ.ngs. Because of the

care (and tJ.me) which was taken in thJ.s inJ.tJ.al process of "freezing"

the vortex evolution onto a large sheet of paper, the outlines have

proved J.nvaluable J.n later interpretations of the data.

At an average of 35 pOJ.nts per run the followng data was

recorded for each vortex ring (refer to Fig. 2-5):

,

Section 4-3.1 Experiment 82

Run d Rough

Number D ReT Visualization

1 0.10 5000 dye

2 0.10 4000 dye

3 0.10 3000 dye

4 0.15 5000 dye

5 0.15 5000 H2 6 0.15 4000 dye

7 0.15 4000 H2 8 0.15 4000 H2 9 0.15 3000 dye

10 0.15 3000 H2 11 0.25 5000 dye

12 0.25 5000 H2 13 0.25 3000 dye

14 0.32 5000 dye

15 0.32 4000 dye

16 0.32 3000 dye

17 0.42 5000 dye

18 0.42 5000 H2 19 0.42 4000 dye

20 0.48 5000 dye

21 0.48 5000 H2 22 0.48 4000 dye

23 0.48 3000 dye

24 0.64 5000 dye

25 0.64 4000 dye

26 Irregular orifice dye

TABLE 4-1

General identification for experimental vortex ring runs

Sect~on 4-3.1 Exper~ment 83

x(t)

R(t)

The d~stance from the orif~ce; ~.e., the traJectory.

The rad~us to the core centerline. Th~s w~s obta~ned

accurately for the vortex r~ngs v~sual~zed w~th hydrogen

bubbles, and for three rings v~sualized w~th dye. The

others were est~mated.

R'(t) The radius of the vortex ring to the outermost streaml~ne.

T(t)

From the dyed r~ngs, th~s ~s taken to be the edge of the

dyed region.

The thickness of the vortex r~ng. Also obta~ned from the

dye visualizat~ons, th~s measurement was generally less

accurate than that for R' due to the presence of a wake.

In~t~ally the data was reduced completely by hand [4J, but when the

PDP-1 1 /45 computer was made ava~lable for my use all the data was

stored on per~pheral d~sk back storage and subsequent data reduction

handled v~a algol programs.

CORRECTIONS. For vortex rings formed wlth the smaller or~f~ces

~t ~s often the case that the piston stops eJect~ng water wh~le the

vortex r~ng ~s travel~ng up the tube. The traJector~es for these

rings were therefore corrected by subtracting the piston traJectory

from the vortex r~ng trajectory. That ~s:

x(t) - x (t) x(t)~{ x( t) - i

p

t < T P

t > T P

(4-4)

where T and X are the plston travel t~me and stroke length p p

respect~vely. In add~tion ~t was necessary to correct the radial

values for the effect of refract~on. Th~s was made d~ff~cult by the

presence of three substances w~th dlfferent refract~ve ~nd~ces: a

cyl~ndr~cal column of water, a 1 em. th~ck plex~glass tube, and the

surrounding a~r. Deflning P A as the apparent rad~us as seen from

outs~de the tube and p as the real radius, a short algorithm was

developed to calculate p/Po as a function of PA/Po . The result is

FIGURE 4-5

Refraction corrections: apparent vs. real radius.

Section 4-3.1 Exper1ment 85

plotted in Fig. 4-5. All values of Rand RI were corrected for

refract10n w1th th1S algor1thm.

4-3.2 Vortex R1ng Velocity

The traJectory of each of the 26 runs was plotted uS1ng 11near,

semi-log, and log-log scales. It was d1scovered that for many runs

one, two, or three linear per1ods, or "reg1mes," can be read11y

identif1ed on one of the three traJectory plots. In each regime the

properly plotted trajectory can be f1t with a straight 11ne, the slope

of the stra1ght line changing from one regime to the next over a

relat1vely short period of t1me.

DATA ANALYSIS. Havlng made the 1n1tial discovery that for some

vortex rings the time dependent form of the r1ng traJectory (and thus

veloc1ty) can be identified w1th1n reg1mes, an extremely careful

analys1s was undertaken to obJectively search for and 1dent1fy such

reg1mes 1n all the runs, and to accurately determlne the slopes and

1ntercepts of each linear reg1me.

One approach used was to compute and plot, uS1ng each of the

three scales, the veloc1ty d1stributlon for each run. Between each

two data p01nts the slope, or "veloc1ty" was computed. The velocity

range was then d1vided 1nto b1ns and the number of slopes in each b1n

determ1ned and plotted. Ideally, a traJectory with three 11near

reg1mes, for example, would have three peaks in the veloc1ty

d1str1bution, and as the number of b1ns 1S 1ncreased, each of these

peaks should split into two or more due to osc111at1ons about the

mean. Of course, the noise level should go up as well. An example 1S

shown in Fig. 4-6 for Run 9. Using a 11near scale, three regimes can

be clearly identified, resulting 1n three broad peaks in the veloc1ty

dlstribut10n w1th 10 b1ns. Increasing the number of b1ns to 15, the

three peaks are better def1ned and begln to split. W1th 30 blns each

major peak has split lnto two smaller peaks. In general, however,

Section 4-3.2

o ~ __ 4-__ ~ __ -L __ -L __ ~ __ ~~ __ ~ __ ~ __ _

6~~~-r~~--r-~-.-r~~r-r-'-~

oL-Z-~-L~~---L-~~~~~~~~-

5 ~~~~~rTTO~~'-rrTl~rT~rr~~

bin number

FIGURE 4-6

The velocity distribution for Run 9.

Experiment 86

10 bins

15 bins

30 bins

Sect10n 4-3.2 Exper1ment87

th1s method d1d not work well. Due to the low number of p01nts 1n

each bin to beg1n with, the n01se level is usually too h1gh to obtain

the quant1tative information des1red.

A more fru1tful approach to objectively 1dent1fy these linear

regimes 1S to choose a p01nt near the beginning of a reg1me, and

compute the slope from that data p01nt to each succeSS1ve p01nt.

Numbering the data p01nts 1, 2, 3, ... , and plotting the slope from

the first p01nt to each succeSS1ve point vs. the p01nt number, we

would expect that 1f a 11near reg10n eX1sts, the plot would osc111ate

at f1rst, but soon settle down around the mean slope of that reg1me.

The end of the regime would be 1dentified as the location where the

plot begins to move away from the mean slope. Having ident1f1ed the

end of a reg1me, one can then choose a fixed p01nt near this end, and

repeat the procedure in the Oppos1te d1rect10n 1n order to 1dent1fy

the beg1nning of the regime.

This techn1que worked quite well, w1th two mod1f1cations.

F1rst, rather than compute the d1rect slope to succeS1ve points, the

least squares fit to a straight 11ne was computed. The result1ng plot

can be expected to dampen more rap1dly to the mean value desired. In

add1t10n, the mean value for the slope of a regime 1S now obta1ned

with a least squares fit over all the data p01nts w1thin that reg1me.

The second mod1f1cation was to compute the average res1duals along

with the least squares fit, and plot them also against point number.

When the end of a regime is encountered, the value of the residual

1ncreases sharply. The result 1S an excellent 1nd1cat10n of the

beginn1ng or end of a regime.

Using Run 9 again as an example, a plot of least squares f1t

slopes and res1duals from point 9 (the beginning of the f1rst

reg1me) to succeeding points is shown 1n Fig. 4-7. The end of the

first reg1me 1S clearly ident1fied from the plot of the res1duals as

p01nt 20, and the slope is seen to oscillate but soon settle down

around the value 83.7.

Section 4-3.2

89

Q) Po. o

...-i CJ)

point number

FIGURE 4-7

The least squares fit slopes and residuals vs. point number for Run 9.

Experiment 88

Sect10n 4-3.2 Exper1ment 89

RESULTS. The result of this analysis is that one, two, or three

linear reg1mes can be ident1f1ed for all our experimentally produced

vortex rings (except for Run 8; see Sect10n 4-4.3). We f1nd that

based on the scale used, the vortex r1ngs can be divided 1nto three

groups wh1ch parallel the three class1f1cations discussed

qual1tat1vely 1n the last sect10n. These groups are:

Smallest or1f1ces/H1ghest values of Jet and r1ng Reynolds number:

x(t) ~ In t (semi-log scale).

2. Intermed1ate orif1ces/Intermediate values of Jet and r1ng Reynolds

number:

x(t) ~ t (11near scale)

3. Largest Orif1ces/Lowest values of Jet and ring Reynolds number:

In x(t) ~ In t (log-log scale).

The slopes for each reg1me for the 26 runs are g1ven in Table A-2 of

Appendix A.

The observation that regimes can be ident1f1ed 1n the traJetory

has signif1cance w1th respect to the veloc1ty of the vortex r1ng.

What we have found 1S that at all Reynolds numbers vortex r1ngs

propagate through reg1mes where the t1me dependent form of the r1ng

velocity can be ident1fied, two reg1mes being separated by a sudden

change in the velocity of the vortex ring. In addit10n, we find that

the form of time dependence can be grouped by Reynolds number as

follows:

1. High Reynolds number, turbulent vortex rings:

U = A

n t

where A > A 1 n n-,

2. Intermed1ate Reynolds number, laminar vortex r1ngs:

U = B where B a n n-1

(4-5a)

(4-5b)

(4-5c)


The subscrl.pt n indl.cates a regl.me; the values A , B ,and a are n n n constants within a regl.me but dl.fferent from one to the next. For

example, a vortex rl.ng of "l.ntermediate" Reynolds number travels wl.th

a constant mean velocity for a perl.od of time, goes through a rapl.d

change l.nvolving a decrease l.n speed, and contl.nues l.nto the next

regl.me again Wl.th a constant velocl.ty, but slower.

Consistent wl.th the description of "transitl.onal" vortex rings

given l.n the last section, we find that the transitl.on in the time

dependent form of ring velocl.ty as you move from one group to the next

does not happen abruptly. For example, the fl.rst two regl.mes of Run 4

are best represented with a semi-log scale but the third regime with a

linear scale, indl.catl.ng a transitl.on from the turbulent to the

laml.nar group. Between the intermedl.ate and low Reynolds numbers is

Run 22, where a ll.near scale l.S best in the first regime, but a

log-log scale l.n the second.

One addl.tl.onal note should be made with respect to repeatability

of these results. After reducing the data for Runs 9 (dye) and 10

(hydrogen bubble) l.t was dl.scovered that the values for T ,X and p p

therefore ReT and Re j were nearly the same. One would hope therefore,

that the traJectories of the two vortex rl.ngs would be close to one

another. Indeed, as l.S shown l.n Fl.g. 4-8, when plotted one on top of

the other there is very ll.ttle difference between the two. Thl.S

observation gl.ves us confldence l.n the stabl.lity of our experl.mental

procedure.

4-3.3 Vortex Ring Radius and Thl.ckness

Along Wl.th the trajectory, the radl.l. Rand R' and the thickness

T of the vortex rings were measured. Radl.al values were corrected for

refractl.on from the calculatl.ons presented l.n Fl.g. 4-5. Values of R'

and T were obtained from the vortex rings vl.sualized Wl.th dye, and R

from those vl.sualized wl.th hydrogen bubbles. Accurate values of R

were also obtalned from dye vl.suall.zatl.on measurements in Runs 13 and

Section 4-3.3

rrrT-rrrTTTTl ~

0

~

0

c

0

• 0

c

0

., 0

e

0

6

0

t> 0

+0

0

t> >-t!j 0

I 6

0

~

~ 0 ... +0 0:

~ 0

0 >

I I ~ 0 I() 0 I() (I') (I') N N

(WO).JX

FIGURE 4-8

Repeatability: comparison in trajectory between Runs 9 and 10

circle: run 9 cross: run 10

Experiment 91

0 I() .

,..., 0 CD <I)

'"' CD § ...

·0

0

• 0

0

1> 0

·0

• 0

·0

·0

·0

·0

·0

·0

• 0

0 I()

Sect10n 4-3.3 Expenment 92

15, and fa1rly accurate values in Run 14. W1th these runs clear

water, wh1ch was entrained upon formation of the vortex ring, marked

the center of the core.

The volume of the vortex ring was computed from T and R' using

the formula for an ellipsoid,

(4-6 )

in order to compare with the volume of flu1d ejected from the orifice

V •• eJ

The rat10s R'/po, R/po, T/po, V/V , and T/R' were computed and eJ plotted as a funct10n of time for each run. These plots are given

also 1n Append1x A, F1gS. A-1 through A-26. The mean of R'/po and

T/R' are approx1mated with stra1ght l1nes calculated by a least

squares fit.

Using these plots together with the traced 1mages of the vortex

rings, a formation t1me 1S determ1ned for each run. The vortex ring

1S estimated to have completed its initial format1on process and to be

independent of the eJect10n at th1s time. A list of these initial

t1mes to and the result1ng in1t1al r1ng Reynolds numbers for each run

1S given in Table B-1 of Appendix B.

4-4 DISCUSSION

4-4.1 Reynolds Number Dependence

It was found that the three classes of vortex r1ngs wh1ch are

1dent1f1ed by the time dependent form of veloc1ty can be grouped by

Reynolds number. In Section 2-3.2 we p01nted out that thevelocity of

a vortex r1ng 1S d1rectly proportional to the strength and inversely

proport1onal to the rad1us of the vortex ring. That is,

r 'V UR (4-7)


A change 1n veloc1ty, therefore, 1S an 1nd1cat10n of changes 1n the

total circulat10n of the vortex r1ng Wh1Ch result fromdiffus10n of

vort1c1ty from the core and convection 1nto the wake.

HIGH REYNOLDS NUMBERS. The vortex rings at very h1gh Reynolds

numbers are found to be dom1nated by small scale turbulent motions.

Qual1tative observat10ns of these rings suggest that vort1city 13

constantly be1ng eJected from the vortex r1ng 1nto a rather heavy

wake, apparently a result of turbulent diffus10n from the core (see

photograph in F1g. 3-4 and draw1ng in Fig. 2-7b). This be1ng the

case, one would expect from Eq. (4-7) a decrease in veloc1ty

cons1stent with a rap1d loss of strength from turbulent diffusion and

convect10n of vorticity.

From the traJectory plots we f1nd that for ReR ~ 20000 - 30000

and above the decrease 1n vortex r1ng veloc1ty goes 1nversely with

t1me. From Eq. (4-5a), U = A It. A 1ncreases from one regime to the n n

next, suggest1ng a possible 1ncrease 1n veloc1tYi however, the slope

analys1s 1nd1cates that the veloc1ty at best rema1ns constant dur1ng

this short trans1t10nal per10d, and might decrease slightly.

Both Maxworthy [34] and Krutzsch [28] have reported a 1ft

veloc1ty dependence for vortex r1ngs at h1gh Reynold~ numbers, as well

as a trans1tlonal per10d where the veloc1ty rap1dly changes. The1r

data, however, 1nd1cates a decrease 1n the slope A rather than an n

1ncrease. The reason for the d13crepency 1S unclear.

LOW REYNOLDS NUMBERS. At the Oppos1te extreme are the vortex

rings at very low Reynolds numbers Wh1Ch are dominated by V1SCOS1ty.

The observance of a laminar wake suggests a loss of vort1city from the

vortex r1ng resulting from viscous d1ffus10n of vort1c1ty from the

core and convection 1nto the wake. We would expect, therefore, a

decrease 1n total c1rculation and velocity cons1stent w1th the much

slower process of molecular d1ffus10n.

Our exper1mental results 1ndicate that for ReR ~ 1000 - 2000 and


below, a much less rapid decrease in veloc1ty is observed. From Eq.

(4-5c) :

U '" 1 a.

t n

(4-5c)

where a. 1ncreases from one reg1me to the next and has values '" 0.13 n

to 0.27. Th1S result 1S consistent with the models for V1SCOUS vortex

rings discussed in Section 2-4.2. For the extreme of a thin, viscous

core of vorticity the velocity decreases more slowly than (4-5c), like

-In t. On the other hand, in the asymptot1c limit of large times,

when the core has the maX1mum th1ckness possible, the velocity

decrease is more rapid than (4-5c), like t-3/2 . The measured rate of

velocity decrease is cons1stent with these models since the Slze of

the core for experlmentally produced vortex rings is generally

somewhere between these two mathematical extremes. It is interestlng

that both the asymptotlc and the measured velocities have a power law

form. The observatlon that a. lncreases from one regime to the next n

mlght suggest that for large times a. -+ 3/2. n

INTERMEDIATE REYNOLDS NUMBERS. We have found that for a very

wlde range 0 f Reynolds numbers ReR '" 2000 to 20000, the vortex rlngs

propagate through regimes of constant velocity. That lS,

U = B n

where B decreases from one reglme to the next. These vortex rings n

tend to be well formed with little or no apparent wake, indicating

little loss of vortlclty from the ring, and therefore little change in

ring strength; an example lS shown in the photograph just before the

abstract. Whereas the vortex rings at high Reynolds numbers are very

turbulent in character, and those at low Reynolds numbers are viscous,

the vortex rings in the intermediate Reynolds number range exhlblt a

basically inviscid character. It appears that the Reynolds number is

high enough for these rings that viscosity lS not important, yet low

Sectlon 4-4.1 Experlment 95

enough that turbulence has not developed.

4-4.2 Periods of Rapid Change: Instabilit1es

Between each reglme is a short per10d of rap1d change 1n

veloc1ty, suggest1ng sudden changes ln the structure of the vortex

r1ng. It lS natural to suspect that these sudden changes 1n veloc1ty

are related 1n some way to the development of instab111t1es along the

vortex core. There 1S some eV1dence to support this V1ew.

Waves along the vortex core result in osc111ations 1n the

measured vortex r1ng radius about a mean (see f1gures in Append1x A).

It 1S reasonable to suppose that a breakdown in the structure of the

core, say by the breaking of unstable waves [36], would man1fest

1tself as unusually large osc1llat1ons, or a sudden change 1n the r1ng

rad1US. As the core reorgan1zes from such a breakdown, f1n1te

amounts, or chunks, of vort1cal flu1d m1ght be ripped from the core,

convected to the outermost reg10n of the vortex r1ng, and f1nally

depos1ted 1nto a wake. Such a sudden loss of a fin1te amount of

vort1c1ty would result in a sudden loss of c1rculat10n, and therefore

a rap1d decrease 1n ring veloc1ty. We m1ght also expect that as a

vort1cal chunk 1S convected around the core 1t would eventually show

up as a bulge in the outermost dyed streaml1ne, moving around the

vortex ring 1nto the wake--at some t1me later than the occurrence of

the decrease 1n velocity.

Bulges follow1ng the pattern Just described have been observed

1n several of our vortex rings, result1ng 1n an lncrease in RI as the

bulge moves around the r1ng 1nto the wake (see, for example, F1g.

A-14b, Append1x A). L1kewise, one can observe 1n the data large

osc1llations and sudden increases in Rand RI at t1mes which appear to

be correlated with the periods of rapid veloc1ty change. From the

hydrogen bubble visual1zations one observes as well periods where the

vortex core suddenly contorts into an S shape.

In order to test for a possible correlation, all of these

Section 4-4.2

10

0.1 0.1

FIGURE 4-9

period of sudden velocity changes

Correlation of sudden velocity changes with instabilities.

Experiment 96


periods of apparent 1nstab1lity were recorded and plotted against the

nearest period of sudden change 1n vortex ring veloc1ty. Th1s plot is

shown 1n F1g. 4-9. Many of the reg10ns of 1nstab1lity correlate well

w1th the reg10ns of sudden velocity change. There 1S certainly a

suggest10n that the sudden changes 1n the velocity of the vortex r1ng

are related in some way to the development of instabilit1es along the

vortex core.

4-4.3 Additional Observations

In several runs other observat10ns were made wh1ch should be

noted.

ROCKING INSTABILITY. We d1scussed in Sect10n 4-2 the

trans1t1onal Reynolds numbers wh1ch appear to be character1zed 1n part

by the occurrence of unusually large-ampl1tude osc11lat10ns. An

example 1S the vortex r1ng of Run 5, representing a trans1t10n between

the high Reynolds number, turbulent vortex r1ngs, and the intermediate

Reynolds number, 1nv1sc1d vortex rings. This vortex r1ng was

bas~cally turbulent ~n character, but exh~bited an ~nterest~ng

oscillat10n. From hydrogen bubble visua11zat10ns, we observe that

early in 1ts 11fe the vortex core suddenly became contorted, mark1ng

the init1at1on of a two cycle oscillation, or rock1ng motion, as

1nd1cated schematically in F1g. 4-10. Although the vortex r1ng

traveled straight up the tube 1t osc1llated through two changes in

or1entation. Accompanying these changes, the radius of the vortex

r1ng was observed to 1ncrease and then decrease (see Fig. A-5b).

S1nce U ~ ~R we observe as well an 1ncrease and decrease 1n the mean

veloc1ty of the vortex ring around the general 1/t decrease

characterist1c of turbulent rings. This large amplitude osc1llation

1n the mean veloc1ty is shown in Fig. 4-11. The p01nts represent the

velocity computed directly between data values, and the solid curve is

the 1/t dependence about wh1ch the data p01nts oscillate. The


1st cycle 2nd cycle

FIGURE 4-10

The rocking oscillation observed in Run 5. The change in radius has been exaggerated for clarity.

time

run 5

~

o 2nd cycle

-c •

'--

::>:, -+-> I-

I \: 0 • '"" ...

0 60

'~"'I r-t

\ I Q)

'4 > 40

20' , , I I I I I I I I I I I I I I I I I I I l I I I I I I

o. 1 0 0.20 0 • 30 0.40 0 • 50 t (me '(sec)

FIGURE 4-11 Rocking Oscillation: velocity plot for Run 5

tJ)

CD n rt 1-" 0 ::s .j::-

I ~ . w

t>:l

.a CD Ii 1-" a CD ::s rt

\0 \0

Sectl.on 4-4.3 Experl.ment 100

two-cycle oscl.llatl.on 1n veloc1ty shows up as smooth, large-ampl1tude

deviatl.ons from the mean traJectory curve of Fig. A-5a.

TURBULENCE IN THE AMBIENT FLUID. In general, the 26 runs for

which data was reduced were chosen from sequences where motions in the

ambient flul.d were ml.nimal. Run 8, however, was chosen because the

turbulence level in the water through which the ring would propagate

was obviously quite h1gh. Slnce Run number 7 has nearly the same

inl.tial condl.tions as Run 8, the two traJectories are compared l.n Fig.

A-8. In1t1ally the traJectories coinc1de, but as the vortex ring of

Run 8 enters the turbulent regl.on, it abruptly slows down. The sudden

1ncrease 1n the turbulence level (or 1n the turbulent V1SCOS1ty)

apparently results l.n a very rapl.d 1ncrease in turbulent dl.ffusion,

and l.n the loss of vort1city from the vortex rl.ng. The subsequent

decrease l.n strength and impulse, then, appears as a rapid decrease in

the velocl.ty of the vortex ring.

VISCOUS INTERACTION WITH THE TUBE WALL. In Section 4-2 we

dl.scussed the l.nteract1on of vortex rl.ngs wl.th the tube wall for cases

where the rl.ng Reynolds number 1S hl.gh enough that upon contact with

the tube wall, the vortex rl.ng rapidly breaks down into a turbulent

mass of flul.d. When the Reynolds number is very low, however, changes

l.n ring structure can come about only through molecular motions.

Such was the case with Run 24. After formatl.on the vortex ring

is observed to move towards the tube wall. Upon contact, the core of

the vortex rl.ng nearest the tube wall l.S observed to grow very large

1n size, apparently a result of viscous productl.on of vortl.city at the

tube wall whl.ch l.S of Opposl.te sl.gn to that l.n the core. In this run,

however, the vortex rl.ng d1d not completely break down at the tube

wall, but reformed and proceeded to move towards the opposite sl.de of

the tube.

Espec1ally slgnif1cant is that in both the period before the

vortex r1ng made contact with the wall, and after 1t reformed and

Sect~on 4-4.3 Experiment 101

moved towards the oppos~te wall, the veloc~ty exh~b~ted a power law

t~me dependence (see F~g. A-24). Th~s observat~on lends support to

the suppos~tion that a power law ~s character~stic of the velocity of

vortex rings at very low Reynolds numbers, and therefore

characteristic of the rate of loss of c~rculat~on due to v~scous

diffusion. In add~tion, s~nce this vortex r~ng clearly went through a

breakdown and reformation process as ~t ~nteracted with the tube wall,

we m~ght conclude that the rap~d changes 1n vortex ring veloc~ty

observed ~n general between reg~mes are also due to a breakdown and

reformation of the structure of the vortex ring.

IRREGULAR OCCLUSION. Along s~m1lar l~nes are the results for a

turbulent vortex ring formed from a very irregular or1f~ce (Run 26).

W~th appl1cat10n to arter~al stenoses (see Section 5-4), clay was

appl~ed to the tube wall around an orif~ce, creating an extremely

~rregular occlusion. The jet Reynolds number was ~ 44900; a

turbulent, 1ncoherent vortex r1ng was observed to travel up the tube

and collide w1th the tube wall always at the same location. As shown

in F~g. A-26, even for this very 1rregular vortex ring a 1ft

dependence in veloc~ty is observed w~th a sudden change in speed at

one, and possible two points in its evolut1on. This lends support to

a 1ft dependence in velocity as characterist1c of vortex rings which

are dom1nated by turbulent diffusion.

5 Combi11atio11 of

Theory with Experime11t

VucaJL.tu' VoJt:te.x SyUem

5-1 TOTAL CIRCULATION AS A FUNCTION OF TIME

In Chapter 3 we developed the mathemat1cs to kinemat1cally

relate the total c1rculat1on of a vortex r1ng 1n a tUbe to 1tS speed,

slze, and shape. In th1S chapter we use these kinemat1c relat10nships

to compute the total circulat10n as a funct10n of t1me for our

exper1mentally produced vortex rings. The concepts beh1nd this

calculat10n are d1scussed at length 1n Section 2-5.

COMPUTATION. The bas1c kinematic relat10nsh1ps were derived 1n

Sect10n 3-4.1, where we related the follow1ng non-d1mens1onal

parameters:

A U R U = rJ 4nR' E: = Po

A

U lS further d1v1ded 1nto Uo

R' T and R or R

(5-1)

U , the velocity of the vortex ring 1f 1

1t were 1n an unbounded flow less the veloc1ty 1nduced by the presence '" of the tube. U, a funct10n only of E:, lS given by Eq. <3-32). Uo 1

lS paramater1zed uS1ng the core parameter aiR as def1ned by the

mod1fied Kelv1n/Lamb formula*:

A 8R 1 Uo = In a -"3 (5-2)

*For a detailed descr1ption of the use of aiR see Section 3-4.1.

102

Section 5-1 Comb1nat10n of Theory w1th Exper1ment 103

Since the relationships among the parameters 1n (5-1) were

computed totally from k1nematic cons1derat10ns, they apply at any

instant in time where the shape of a vortex r1ng is well represented

by concentrating its total circulation along a c1rcular filament. As

we d1scussed in Section 2-5, such a representation 1nvolves two

approx1mat10ns. The f1rst, concern1ng the effect of a finite core on

the shape of a vortex ring, was addressed 1n Section 3-4.2, and the

second, which concerns the effect of a wake, will be d1scussed 1n

Sect10n 5-2.1.

Because when uS1ng dye v1sualizat10ns, measurements of R' are

generally more accurate than measurements of R, we recast the

relationsh1ps 1nvolving R 1nto more useful relationships 1nvolv1ng R'.

In Figs. 3-11 ~nd 3-12 are shown the calculations for T/R and R'/R as

a function of aiR for d1fferent values of E. Comb1n1ng these

calculat10ns we compute T/R' as a function of R'/po for d1fferent

values of aiR, as shown 1n F1g. 5- 1 • R'/po: 0 corresponds to the

unbounded vortex ring. As we noted earl1er, we see that the th1ckness

(and volume) of a vortex r1ng 1ncreases with increas1ng aiR, or

decreas1ng veloc1ty; for R'/po ~ 0.4 there 1S relatively l1ttle

influence from the tube wall on the shape of the vortex r1ng.

The calculat10ns represented by F1gS. 5- 1 and 3-12 are in a form

sU1tabie for calculat1ng the total c1rculation from flow v1sual1zat10n

measurements. At any 1nstant in t1me, we compute r as follows:

1. A flow visualization measurement of T/R' and R'/po defines a p01nt

on F1g. 5-1. aiR is determ1ned by 1nterpolat1ng between the

constant aiR curves.

2.

3.

4.

Knowing aiR and R'/po, E

wh1ch led to F1g. 3-12.

1S determ1ned from the calculat10ns

A A

Know1ng aiR and E, UO 1S calculated from Eq. (5-2) and U from Eq. 1 A

(3-32). Subtracting the two we f1nd U.

Hav1ng computed U and R (:EPo), we see from Eq. (5-1) that a flow

v1sual1zation measurement of U Y1elds the total c1rculat10n, r.

.. a::: ....... t-

Section 5-1 Combination of Theory with Exper1ment 104

1.00~--~--~--~--~--~------~------~--~

0.80

0.60

--o • 40t:===::::;:::==:::;:==~=------=

O.OO~--~--~--~--~--~--~--~--~--~--~ 0.00 0.20 0.40 0.60

R' I Yo

FIGURE 5-1 The variation of T/R' with R'/p

o for different

"-

values of U . o

0.80

Beginning with the lowest curve, a/R = 0.04,0.05, ••. , 0.50

1.00

Sect10n 5- 1 Comb1nation of Theory with Exper1ment 105

We have calculated not only the strength of the vortex r1ng, but also

its radius, R from measurements of RI, T, and U. Th1s algor1thm is

used with our flow visualizat10n measurements to compute the change in

total circulation as the vortex r1ngs travel up the tube.

As discussed in the last chapter, the trajectory, radius, and

thickness of our exper1mentally produced vortex r1ngs are well

represented by a mean motion plus oscillations about the mean wh1ch

result from instab111t1es and waves along the vortex core. Since 1t

is the mean mot10n with wh1ch we are 1nterested, smooth curves are

passed through the data to compute r. The analysis of the mean trajectory was d1scussed in the last

chapter. We d1scovered the existence of reg1mes for all our vortex

r1ngs, where the t1me dependent form of the velocity 1S given by EQs.

(4-5). The vortex r1ng strength 1S computed uS1ng the values w1thin

each regime as given in Table A-2. A discont1nuity 1n r will

therefore occur at the (art1f1cial) points of discont1nu1ty 1n

veloc1ty between reg1mes.

A least squares f1t to straight 11ne segments 1S used as well to

approx1mate the mean values of RI/PO and T/RI. As shown 1n Append1x A

by the dashed curves, in most cases a stra1ght l1ne is a reasonable

f1t to the data. As a result, however, small discontinuities are

necessar1ly 1nserted into the calculation for r .

RESULTS. The total circulation is calculated as a function of

time for 17 runs 1n wh1ch the init1al ring Reynolds number ReR ranged

from 690 to 50 100. These results, plotted along with the velocity of

the vortex r1ng for comparison, are presented 1n Appendix B, Figs. B-1

through B-17. The velocity 1S 1ndicated with dashed curves, the

circulat10n with solid curves. In order to compare the velocity and

strength in a reasonable way, the scales are adjusted so that each

tick mark represents roughly 10% of the mean value.

It should be remembered that the mean values of the vortex ring

data have been f1tted with piecew1se smooth functions, so that

Section 5-1 Combination of Theory w1th Exper1ment 106

d1scontinuit1es result 1n the plots of velocity and strength. In

reality, of course, these changes occur smoothly, although over a

short per10d of t1me. US1ng the least squares ana1ys1s as a gU1de,

approximate transitions over these short per10ds have been 1ndicated

with smooth, dotted curves.

FORMATION. As we expect from Eq. (4-7) and as discussed in

Sect10n 4-4.1, 1n most cases the strength of a vortex ring as it

propagates up the tube follows the velocity of the vortex ring.

However, some differences are apparent. In many cases, even though

the veloc1ty 1S constant or decreasng at the beginning of a run, the

strength 1S 1ncreas1ng. Careful examination of the plots in Append1x

A and the original outl1nes made from the mOV1es ind1cates that the

vortex ring 1S forming during th1s period and vortic1ty is st1ll be1ng

concentrated 1n the core. Thus, the strength of the ring increases

dur1ng th1s format1on process as suggested by the plots.

UNCERTAINTY. The uncerta1nty 1n these plots is very difficult

to assess. In order to get a feel for the extent of the osc1llat10ns

about the mean, the veloc1ty and strength were computed and plotted

p01nt by point. Th1s 1nvolves the calculation of derivatives by

tak1ng d1fferences between data p01nts, resulting in deviations from

the mean which are very large and unreal1stic. Indeed, this is

perhaps the pr1mary reason for f1tt1ng the vortex ring trajectory with

a smooth curve before taking derivatives to compute the r1ng veloc1ty.

Calculating the velocity and strength between every two data pOints,

however, prov1des a check for the calculations in which mean values of

the vortex ring parameters are used.

An example calculation is shown 1n F1g. 5-2 for Run 1. The

p01nts and dashed l1nes represent the values computed d1rectly from

th~ data, and the sol1d curve represents the mean computed by f1tt1ng

the data with piecewise smooth curves. We see from F1g. 5-2 that the

trans1tion reg10n, which appears as a discont1nuity when the mean

260

.....--o 220 Q) CI)

.......... E 0

-.,. 180

» -+--> .-0 o 140

r-I Q)

>

100

0.10

FIGURE 5-2

run 1

0.15 time

0.20 (sec)

0.25

Comparison of mean velocity and strength with point by ~oint calculations for Run 1. (a) velocity

Ul (!) n rt .... o ::l VI I I-'

C')

~ .... ::l III rt f-J. o ::l

o H'l

~ (!) o Ii '< ~ f-J. rt ::r'

t%j

.a (!) Ii f-J. S (!) ::l rt

I-' o ......

-o 900 Q) CI)

"C\J

E 760 o

-.,.

..c.

....., 620 Q)

c Q)

L ....., 480 CI)

;

I I

I I

I I

I

run

I ',-.... --.-/ ,,~ I ,,'- ~I

1

/e.. \

~ \ \ -- \

--~~ ~--, ------. , ,-, ,-

"r"

340' I 0.10

FIGURE 5-2

0.15 0.20 0.25 t t me (sec)

Comparison of mean velocity and strength with point by point calculations for Run 1. (b) strength

CIl CD n rot 1-" 0 ::l

V1 I ....

("') o ~ 1-" ::l III rot 1-" o ::l

o Hl

rot ::r' CD o Ii '< ~ 1-" rot ::r' t%j

~ CD Ii 1-" EI CD ::l rt

...... o 00

Sect10n 5- 1 Comb1nation of Theory w1th Exper1ment 109

curves are used, 1S shown as a smooth but rap1d change when the

velocity and strength are computed p01nt by p01nt.

5-2 ACCURACY OF THE COMPUTATION METHOD

5-2.1 The Effect of Wake Vorticity on R1ng Shape

The kinemat1c model wh1ch 1S used to compute the circulat10n 1S

based on two approx1mat10ns (Section 2-5). The effect of a finite

core on the shape of a vortex ring was assessed in Section 3-4.2 and

found to be of little consequence. We have yet to assess the

importance of vorticity in the wake on the shape of a vortex ring.

Clearly any such effect will be most important for the r1ngs

w1th the heaviest wakes, so we cons1der the experimentally produced

vortex r1ngs wh1ch have the h1ghest (Run 1) and lowest (Run 25)

Reynolds numbers. Taking the d1fference of the initial and f1nal

strengths from Figs. B-1 and B-17, and div1ding by the distance the

vortex ring has traveled, we f1nd that for Run 1 the average strength 2 2 per un1t length 1S ~ 12.3 cm Isec/cm and for Run 25 ~0.15 cm Isec/cm.

If we now divide by the average total circulat10n of each vortex r1ng

we find that, on the average:

strength 1n 1 cm. of wake

strength of vortex ring =

0.018 {

0.013

Run

Run 25

That 1S, for the h1ghest Reynolds number ring there is about 1.8% of

the total circulation in 1 cm. of wake and about 1.3% for the vortex

ring at the lowest Reynolds number. We therefore use Run 1, the worst

case, to test the 1nfluence of a wake on vortex ring shape.

We consider the same parameters which were used to test the

effect of a finite core: E = 0.40 and aiR = 0.25. From the movies we

observe that trailing vort1city tends to be distributed in a reg10n

Section 5-2.1 Combinat~on of Theory with Experiment 110

about half the rad~us of the vortex r~ng. In order to model the wake,

then, we d~stribute the vort~c~ty w~thin th~s region behind the vortex

r~ng over two rows of circular vortex filaments. The strength of each

f~lament is weighted as est~mated for Run 1. Us~ng the linear~ty

of the B~ot-Savart law, the streamline f~eld ~s computed by summ~ng

the ind~v~dual f~elds of the vortex elements.

The streaml~nes for the vortex ring with and w~thout a wake are

g~ven in F~g. 5-3. The vortex elements which model the wake are

ind~cated w~th dots. the effect of the wake ~s to distort the

streamlines behind the vortex r~ng and to cause a slight asymmetry

about the z = 0 plane ~n the shape of the vortex ring. As is shown ~n

F~g. 5-3c, the rear end of the r~ng ~s drawn out slightly resulting in

a sl~ghtly thicker vortex r~ng.

The d~fference ~n thickness due to the presence of the wake, as

compared w~th a wakeless vortex ring, ~s about 3%. This is well

w~th~n the prec~s~on of the exper~mental measurements. Furthermore we

have chosen a vortex r~ng w~th a heavy wake so that for other rings

the effect ~s probably not even as great. We therefore conclude that

the effect of the wake ~s negl~g~ble for typ~cal vortex r~ngs, and the

s~ze and shape can be accurately pred~cted by concentrating the total

circulation of a vortex r~ng along a c~rcle.

5-2.2 Comparisons with Measured and Estimated Values

We would like to test the accuracy of th~s method with other

methods for computing the total c~rculat~on of a vortex ring. To do

this it is necessary to have both flow visual~zation measurements

of T, R', and U as well as an ~ndependent calculat~on for r

MEASUREMENT. Sull~van ~~ [60] have measured r, U, R, and

aiR us~ng a laser doppler velocimeter (LDV) for two vortex rings

produced in air at d~fferent Reynolds numbers. They include a

photograph for one vortex r~ng, that with the lower value of Reynolds

Section 5-2.1 Combination of Theory with Experiment III

FIGURE 5-3

The streamlines for a vortex ring w~th a wake compared with a wakeless vortex ring. a) The wakeless vortex ring.

Section 5-2.1 Combination of Theory with Experiment 112

,(b)

FIGURE 5-3

The streamlines for a vortex ring with a wake compared with a wakeless vortex ring. b) The vortex ring with a wake.

t:;;~F<~:'without a wake

with a wake

(c)

c) Comparison of outermost closed streamlines.

Section 5-2.2 Combination of Theory with Experiment 113

number (Re R = 3040, Rer = 7780), thus allowing compar~son w~th our

method. Measurements w~th the LDV requ~re the repeatable product~on

of vortex rings, so the photograph ~s only one of many r~ngs measured.

By integrat~ng the measured veloc~ties around the core, they

calculated a total c~rculat~on of 1124 cm 2 /sec. Using the photograph

to measure T and RI and the~r measurement for U we calculate with our

method a total circulat~on of 1135 ± 60 cm 2/sec.

ESTIMATES. The init~al strength of our exper~mentally produced

vortex rings can be approx~mated by relat~ng the total circulat~on of

the fluid ejected by the piston to the Jet veloc~ty and orif~ce s~ze.

If you consider a slug of fluid of length L ejected from the orif~ce

w~th velocity u.(z) the total c~rculat~on content of the ejected fluid J

~s g~ven by*: L

r . = f u. dz eJ 0 J

(5-3)

The total c~rculat~on of a newly formed vortex r~ng ~s therefore

approx~mately g~ven by

r(to) = ~(to) r. • eJ

eJ

(5-4)

where Veto) ~s the in~t~al volume of the vortex r~ng and V . is the eJ

volume of flu~d ejected from the orif~ce. Using these expressions,

the in~tial strengths of our vortex r~ngs were est~mated and compared

w~th our calculated values.

In F~g. 5-4 are shown the compar~sons between the values of

total c~rculat~on calculated using the kinematic theory, and the

values measured by Sullivan ~~ and est~mated from Eq. (5-4). The

comparison is felt to be excellent.

*This expression, which is straightforward to derive, agrees with that used by Saffman (55) to estimate the initial circulation of vortex rings. Maxworthy (36), however, includes a factor of 1/2 for reasons which are unclear.

Section 5-2.2 Combination of Theory with Experiment

1000

-0 QJ ~

;j 100 I§' o CJ

10 10

•

FIGURE 5-4

100

r(to) estimated/measured

Comparison of vortex ring strengths computed using

1000

the kinematic theory with estimates and one measurement.

4t estimated from Eq. (5-4) ~ measured by Sullivan ~ al.

114

Section 5-2.2 Combinat10n of Theory with Experiment 115

VORTEX RING RADIUS. Earlier we pointed out that 1n the process

of computing the strength of vortex rings from measurements of T, R',

and U we compute the radius R as well. For most of our dye

visualizations the estimates of R are not accurate enough to warrant

comparisons with the computed values of R. For Runs 13 and 15,

however, we were able to obta1n accurate measurements of both R' and R

due to the presence of clear water along the core centerline. We were

also able to provide a good estimate for R in Run 14, although not as

accurate as for the other two runs. It would be of interest,

therefore, to compare the data for these runs with the mean values of

R computed using the kinematic theory. These compar1sons are shown by

solid curves 1n Figs. A-13b, A-14b, and A-15b of Appendix A. The

comparisons for Runs 13 and 15 are excellent and for Run 14 qU1te

good. It is interesting that in both Runs 13 and 14 the predicted

value for R initially decreases as does the data. We also made the

comparison between the computed values of R from the dye visualization

of Run 9 with the hydrogen bubble measurements for R in Run 10. These

two runs were found from the trajectory and experimental parameters to

be very close to one another. The solid line through the data in F1g.

A-lOb shows the compar1son and once again 1S quite good.

One additional note can be made with regard to the accuracy of

the Kelvin/ Lamb formula, Eq. (2-15). Sul11van's measurements suggest

that the Kelvin/ Lamb formula overestimates the velocity of vortex

r1ngs by as much as 30 to 40%. Thus, even 1f it were possible to

measure aiR US1ng flow visua11zat1on, the express10n cannot be

expected to accurately predict the strength of a vortex r1ng.

5-3 VORTEX RINGS AND THE REYNOLDS NUMBER

REYNOLDS NUMBER RELATIONSHIPS. As discussed in Section 2-4.2,

two definit10ns for the Reynolds number have been commonly used to

Section 5-3 Combinat~on of Theory with Experiment 116

character~ze vortex r~ngs:

Re = 2UR R \l

Re = r r \l

(5-5 )

As we found from Eqs. (2-16) and (~-22b) r 'V UR so we expect ReR and

Re r to be closely related. Hav~ng computed r for our experimentally

produced vortex rings, it would be of interest to compare these two

defin~tions. In Fig. 5-5 we have plotted Re R against Re r for our

newly formed vortex rings. This plot suggests that at the initial

time of formation, to:

(at t = ~ )

Inserting the defin~tions in (5-5) results in:

r ~ 4UR (at t = to)

(5-6 )

(5-7)

The KelvinlLamb formula, Eq. (2-17) suggests that the proportionality

factor between -r and UR is a function of the core size, aiR:

r = 4nUR [ 8R 1 ]-1 tn - --a 3

(5-8)

Using th~s formula together with Eq. (5-7) to est~mate the core

parameter of the newly formed vortex rings we find that

aiR 'V 0.25 (at t = to) (5-9)

It was for this reason that aiR = 0.25 was chosen in our assessment of

the effect of a finite core (Section 3-4.2) and wake (Section 5-2.1)

on the shape of a vortex r~ng.

It would also be of interest to study relationships between the

Reynolds number of vortex rings and a Reynolds number based on the

ejection of fluid from the orifice. We have observed qualitatively

that as the jet Reynolds number increases, so does the ring Reynolds

number. In Fig. 5-6 we have plotted ReR and Re r against Re j on a

log-log scale. We note that at Rej

'V 25000 there is an abrupt change

Section 5-3 Combination of Theory with Experiment

100

C'"l I

0 ...... ~ 10

"""' 0 .j.J

'-' ~

Q) p:::

1 0.1 1.0

FIGURE 5-5

Relationship between ~eR and Rer for newly formed vortex r1ngs.

117

50

Section 5-3 Combination of Theory with Experiment 118

100

10

1 1

•

FIGURE 5-6

........ / ", ....... ,. . .," . , ., ,

' .. , I. .. ' '/

I I.

·i .' . , , ,

10

-3 Re.x10 J

Relationships between ReR

or Rer and Rej

for newly formed vortex rings.

a) ReR

100


100

10

./

.,.... -,.... ..... ~ /

I • I • /

I •• I·

,/ -II -I

/ ./ /

l~----------~----~------------~--~ 1 10

FIGURE 5-6

-3 Re .xlO J

Relationships between ReR

or Rer and Rej

for newly formed vortex rings.

b) Rer

100


in the trend. This suggests transition to turbulence in the Reynolds

number ranges:

Re j 20000 - 30000

ReR 25000 - 35000

Re 50000 - 70000 r

REYNOLDS NUMBER GROUPS. In Section 4-4 we discussed

correlations between the three Reynolds number groups of vortex rings,

and the rate at which vorticity diffuses from the core and is

convected into the wake. These ideas are supported by our

computations of the time rate of change in strength, shown in Appendix

B. The rate of decrease in total circulation from high Reynolds

number, turbulent vortex rings is observed to be much greater than the

rate for low Reynolds number, viscous vortex rings, an indication of a

more rapid rate of diffusion due to turbulent motions as compared with

molecular motions. Most interesting, perhaps, 1S the very wide range

of 1ntermediate Reynolds numbers in which the strength of the vortex

ring changes very little (in a regime), consistent with the absence of

an observable wake. These vortex rings appear to have a basically

inviscid character.

In order to characterize these groups in a concise way consider

the coefficient of drag for a vortex ring defined in the usual way as

D 1S the drag force on the vortex ring, U its speed, and S the frontal

surface area. D and S are given by:

s = 1TR,2

Inserting these values into (5-10) yields

8 T dU 3U2 dt (5-11 )

Section 5-3 Combinat~on of Theory with Exper~ment 121

In the case of a solid sphere drag ~s a result of shear forces

along the sphere's surface and flow separat~on beh~nd the sphere. The

drag force of a vortex r~ng, however, results from the loss of

vortic~ty, strength, and impulse from w~thin the volume of fluid

comprising the vortex ring. W~thin a regime, the h~gh and low

Reynolds number vortex rings w~ll therefore have a value of CD

consistent w~th the rate at which they lose vortic~ty, whereas vortex

r~ngs at intermed~ate Reynolds numbers exh~b~t no drag, and thus have

a zero coefficient of drag.

S~nce it is the mot~on with~n reg~mes which characterizes the

three groups of vortex rings, we choose po~nts in the m~ddle of each

reg~me to compute CD. These values are shown plotted aga~nst Re r ~n

F~g. 5-7. Th~s figure n~cely ~llustrates the three groups of vortex

r~ngs. The high Reynolds number, turbulp.nt vortex r~ngs have the

h~ghest values of CD; the low Reynolds number, viscous rings have

generally lower values of CD; and those vortex r~ngs ~n the

intermed~ate Reynolds number, "inv~scid" group have zero coeff~cient

of drag.

The transition reg~on between these groups is expected to depend

in general not only on the Reynolds number, but also on such factors

as geometry and roughness of the orif~ce, piston speed

character~st~cs, the state of the ambient flu~d into wh~ch the vortex

ring propagates, etc.

5-4 ARTERIAL STENOSES AND THE WALL PRESSURE

A motivation for the study of vortex rings propagating up a tube

is its applicat~on to arter~al stenoses. An arterial stenosis is a

constr~ctiQn in an artery which results from the bu~ldup of

cholesterol along the arterial wall, or from a d~seased heart valve

which will not open completely. As blood is forced through these

20

15

10

CD

5

o

\

000:>\

1

FIGURE 5-7

" "-" "\.

o

o " ......... .......

Rerxl0-3

The drag coefficient of vortex rings plotted against Reynolds Number

10

/0 I o f I

0 1 0 o /0

o /

o / /

/ /

/.

100

Ul (I) n rt .... o ::s U1 I

W

(') 0 g. 1-'-::s III rt ..... 0 ::s 0 I-h

~ (I) 0 I'i '< ~ 1-'-rt ::r ~ .a (I) I'i ..... S (I)

::s rt

I-" N N

Section 5-4 Comb~nation of Theory w~th Experiment 123

constr~ctions, sounds called murmurs are produced. Murmurs can be

heard with a stethoscope, and have been used in a qual~tative way for

years to assess the severity of a stenosis. To determ~ne with

precis~on the degree and locat~on of a stenosis, however, usually

requ~res ~nvest~gative surgury or the use of radioact~ve tracers.

In recent years ~nterest has developed ~n the possibil1ty of

using murmurs in ~ more quant~tative way ~n order to pred~ct w~th more

accuracy the severity and location of a stenosis. Understand~ng the

source of murmurs, however, requ~res an understanding of the

formation, development, and structure of the turbulent flowf~eld which

is produced as blood is forced through the constriction.

Based on the suggestion that the flow of blood through a

stenosis reaches a quas~-steady state at peak systole, studies in the

past have concentrated mainly on steady flow models [65]. As is shown

~n Fig. 5-8, however, the accelerat~on of blood into the aorta ~s

extremely rap~d (",4650 cm/sec 2 ), so that, at least for a valvular

stenos~s, the flow is dom~nated by pulsat~le effects. Farther from

the heart a steady component ~s superposed over the ejection pulse,

but pulsatile effects, espec~ally for h~ghly occluded vessels, are

st1ll expected to play an ~mportant role ~n the development of the

post-stenotic flowfield.

As 1S ind~cated 1n F~g. 5-8 our experiment was des~gned using

parameters relating to the arter~al stenos~s. Because the peak

Reynolds number of blood ejected from the aorta ~s about 5000 we chose

tube Reynolds numbers of roughly 5000, 4000, and 3000, and a piston

travel time of roughly 0.170 seconds to correspond to a single

eject~on of blood from the heart. W~th regard to arter~al stenoses we

are most ~nterested ~n the smallest or~f~ces which correspond to the

highest degree of occlusion. These orifices are associated with the

production of turbulent vortex rings followed by the production of a

turbulent Jet. Entirely a result of the pulsatile nature of the flow,

the vortex r~ng travels up the vessel and collides with the vessel

wall, causing large pressure fluctuations.


o 0.5 1.0

time (sec)

FIGURE 5-8 Speed and Reynolds number of blood ejected from the human heart into the aorta.


IRREGULAR OCCLUSION. In order to visualize the development of

vortex rings with a more physiologically realistic occlusion, an

orifice was made very irregular by applying clay to the tube 1nter10r

to an area reduct10n of roughly 92%. It was found that highly

turbulent, rather incoherent vortex rings could still be readily

identified and observed to travel up the tube, col11ding w1th the tube

wall always at the same locat1on.

WALL PRESSURE. Since the fluctuations of the arterial wall are

heard at the surface of the body as noise we would 11ke to analyse the

pressure at the tube wall due to the passage of a vortex ring, and

compare th1S with an est1mate of the magn1tude of pressure

fluctuations Wh1Ch might result from a collision with the arter1al

wall. The pressure at the tube wall can be calculated from the

unsteady Bernoulli equat10n:

o<P 2 P + ~p u + p

m at m constant (5-12)

U 1S the velocity at the tube wall and <P 1S the unsteady potent1al

funct1on. Since t

<P <p(Po, z - f U(s) ds ) o

we f1nd that the pressure at the tube wall relat1ve to the pressure at

1nf1nity is given by:

P - P = 00

- kp (u2 - 2uU) 2 m (5-13)

where U 1S the velocity of the vortex ring. We wr1te u in the

non-dimensional form:

r u(PQ, z) = 4nR u(po, z)

f' Computat10ns for u, obtained by taking the z der1vative of <P in Eqs.

Section 5-4

2

1

o

Combination of Theory with Experiment 126

o 0.5

z

FIGURE 5-9

The non-dimensional wall velocity as a function of zIp for different values of E.

o

1.0

Sect10n 5-4 Combination of Theory w1th Exper1ment 127

(3- 1 9), (3-23) and (3-27) are given in Fig. 5-9 for different values

of £

In order to compute the pressure at the tube wall as a vortex

ring passes by we need to know r, U, and R as a function of time. U

and R are obtained from exper1mental measurements, and we now have

computations for r. Since we are concerned most with the smallest

occlusions wh1ch produce the most energetic vortex rings, we compute

the pressure as a funct10n of t1me at 1.5 tube d1ameters from the

orifice for Runs 1 and 3. The results are given in Figs. 5-10 and

5-11. Time 1S measured from the formation of the vortex r1ng and the

vert1cal 11ne ind1cates the t1me at which the vortex r1ng passes the

point where the pressure 1S computed. We note that the peak pressure

drop does not necessar1ly c01nc1de w1th the t1me when the vortex ring

is 1.5 diameters downstream, and that the pressure profile is

asymmetr1c. Th1s is because the vortex r1ng 1S both loosing strength

\ and increasng 1n size as it travels up the tube.

The peak pressure change at the tube wall due to the passage of

these turbulent vortex rings is apparently on the order of 1/3 - 1

mbar. It is interesting to note that this 1S the same order of

magnitude as the peak rms wall pressure measured in steady flow models

of arterial stenoses [65].

The pressure resulting from a collision w1th the vessel wall,

however, can be expected to be much higher. Using Eq. (2-14) to

estimate the 1mpulse of the vortex r1ng, one can obtain an order of

magnitude estimate for the pressure imparted to the tube wall due to a

coll1sion by a vortex r1ng. Based on values for the smallest or1fices

one estimates pressures at least one order of magnitude larger than

what 1S shown in Figs. 5-10 and 5- 11 • This would suggest that the

product10n of vortex r1ngs beh1nd arterial stenoses m1ght be 1mportant

w1th respect to flow-induced vibrations at the vessel wall. Indeed,

the repetition 1n col11sions of vortex rings with the vessel wall

behind an arterial stenosis might account for the commonly observed

phenomenon of post-stenot1c dilatation--the weakening of the arter1al

wall behind an arterial stenosis.


,....., L eG

..0 E

Q) L :;, C/) C/) Q) L Q.

time 0.173

(seconds)

FIGURE 5-10

The pressure at the tube wall 1.5 diameters from the orifice for Run 1.

0.187

L C'O

...0 E

Q)

L :J cJ) cJ)

Q)

L a...


0.20 time 0.225 (seconds) 0.331

FIGURE 5-11

The pressure at the tube wall 1.5 tube diameters from the orifice for Run 3.

(The) woltld tube lLeplteunt6 an in6-utU.ely c.omplex pILOC.Ulo 06 • • • movement: and development: wluc.1t .{A c.ent:elted .(.n a lLegion ind.tc.iUed by the boundtvu..u 06 the tube. f{owevelt, even ouU.(.de the tube, eac.h 'paItUci.e' ha6 a 6.(.eld th.a.t extend6 tlVtough lopac.e and meltgu Wdlt the 6.(.eld6 06 othelt pa.!LUc.lu. A molLe viv.(.d .unage 06 the MU 06 tlung thiU i6 meant i6 a660ILded by c.otv...(.de/t.(.ng wave 601tm6 M vouex lotlwc.:tultU in a 6loWtng 6t1team • • • The 6low pa..tteltnlo meltge and un.t.te, .(.It one whole movement: 06 the 6loWtng lotlteam. Thelte.{A no 6hMp d.tv.{Awn be-tween them, nOll Me they to be llegMded M II epMiUely OIl .(.Itdependently exilotent ent:d.tu.

-61tOm V. Bohm, "Fltagmen.:ta;t(.on and Wholenulo"

6 Summary

As is indicated by the title of this report, kinematic and

dynamic aspects of vortex ring propagation in a tube have been

discussed. The kinematics is embodied in the mathemat1cs wh1ch was

developed in Chapter 3, and the dynam1cs enters via the flow

visualizat10n measurements d1scussed 1n Chapter 4. Using the purely

kinematic theory together with the data, we computed the change in

vortex r1ng strength as a function of time, a dynam1c result. We

should now like to summarize what we have learned.

KINEMATICS. We exploit two important properties of the

B10t-Savart Law. F1rst, since only conservation of mass for an

incompressible flow is used in its der1vat10n, it 1S a purely

kinemat1c express10n. Whether the flow 1S viscous in character or is

dominated by turbulent fluctuations, if the vorticity field is known

or can be adequately modeled at a g1ven instant in t1me, then the

velocity field can 1n priciple be computed at that t1me from the

Biot-Savart Law. We provide such a model for a vortex ring in a tube

in order to compute the k1nematic relationships among the vortex ring

veloc1ty, radius, thickness, and strength. To the extent to which our

model is accurate in computing these relat10nships, it app11es for all

vortex rings be they viscous in nature, or highly turbulent.

Another important aspect of the Biot-Savart Law is the linearity

of the vorticity field with respect to the velocity field. Making use

130

Chapter 6 Summary 131

of this property we split the vort1city f1eld 1n two parts: the

vortex ring itself, and the wake. We found that effects of the wake

on the shape of the vortex r1ng are at most about 3% and can be

neglected.

In addition, we show that the effect of a fin1te core on the

vortex r1ng shape 1S slight, so that we can accurately model the

vortex ring by concentrating 1tS total circulat10n along a c1rcular

vortex filament placed aX1symmetrically 1n a tube. In a frame of

reference fixed w1th respect to the tube we f1nd that the outer

flowf1eld of a bounded vortex r1ng, like the unbounded vortex r1ng,

has a dipole character, but w1th more of an axial flow due to the

conf1n1ng nature of the tube (Fig. 3-4]. The wall-induced flow on the

vortex ring is 1n a d1rection opposite to 1tS mot10n [Fig. 3-5] so

that the tube acts to slow the vortex r1ng down. The extent to which

the speed of the vortex r1ng is decreased (as compared with the same

r1ng 1n an unbounded flow) depends on the relat1ve rad1US of the

vortex r1ng to that of the tube (Eq. (3-32), Fig. 3-7].

In a frame of reference mov1ng with the vortex r1ng, the extent

of the vortex ring, that 1S, the volume of flu1d mov1ng with the core

1S def1ned. The speed w1th which the vortex ring travels depends on

dynamic cons1derat10ns such as the change 1n size of the core, the

d1str1but10n of vortic1ty, etc.; however, general k1nemat1c

relationships among the r1ng velocity, strength, slze, and shape must

be met at all times. We compute these relat10nships (F1gS. 3-11 and

3- 12], and f1nd that, for a vortex ring with spec1f1ed strength and

size (i.e., radius), the thickness and volume of the vortex r1ng

1ncreases w1th decreas1ng speed. In add1tion, we find that the effect

of the tube 1S to 1ncrease its volume further. Thus, from k1nemat1cs,

we conclude that the effect of a tube on a vortex r1ng is, 1n general,

to decrease 1tS velocity and 1ncrease 1tS volume as compared with the

same vortex r1ng in an unbounded flow.


DYNAMICS. Once the total circulation, rad1us, and veloc1ty of a

vortex ring have been established from the dynamic processes of vortex

formation, convection, and d1ffus10n, the shape of the vortex ring is

specified by the kinematic relationships just described. We therefore

work backwards, and from measurements of the veloc1ty, rad1us, and

thickness of a vortex ring traveling up a tube, determine its total

circulation as a function of time-- a dynamic calculation. It should

be kept in m1nd, however, that the dynam1cs enters through the

experimental measurements of the time rates of change in ring

velocity, th1ckness, and radius--the role of k1nemat1cs 1S to relate

these quantities at any 1nstant in t1me to the total circulation of

the vortex ring.

W1th experimental measurements of vortex rings from very low to

very high Reynolds numbers we f1nd that the vortex r1ngs can be

divided 1nto three groups which are characterized by the rate at wh1ch

total circulation is lost from the r1ng into a wake [Fig. 5-7]. At

very high r1ng Reynolds numbers (Re R 'V 20000 - 30000 and above),

turbulent d1ffusion of vort1c1ty and 1tS subsequent convect10n out of

the vortex ring results in a roughly 1ft decrease 1n r1ng velocity and

strength. On the other hand, at very low Reynolds numbers (rv 1000 -

2000 and below) the much slower process of molecular d1ffusion results

in a less rapid decrease 1n vortex ring velocity and strength, l1ke a

1ft where a'V 0.13 to 0.27 for our exper1mentally measured vortex

rings. There is a very wide range of intermed1ate Reynolds numbers

('V 2000 - 20000), however, where the vortex ring has bas1cally an

inviscid character. Viscous d1ffusion proceeds at such a slow rate

that very 11ttle vort1city is lost from the vortex r1ng, and the total

circulation of the vortex ring remains essent1ally constant.

We also discovered, however, that for all our exper1mentally

produced vortex r1ngs, periods of rapid change in ring velocity and

strength would per10dically occur. That is, the time dependent forms

of ring velocity and strength just described are 1dentif1ed in

reg1mes, two regimes be1ng separated by a rap1d change in ring


velocity and strength. These per10ds of rap1d change are postulated

to result in some way from the growth of unstable waves along the core

of the vortex r1ng which, as they break, m1ght result 1n a per10d1c

reorganization of the core.

Relating to pulsat1le effects in arter1al stenoses, the pressure

at the tube wall 1.5 d1ameters from the orif1ce was calculated for our

smallest occlus1on. It was found that the maximum drop 1n pressure as

these vortex r1ngs travel up the tube 1S on the order of 1/2 mbar, the

same order of magnitude as the peak rms pressure measured 1n steady

flow models. Col11s1ons of vortex r1ngs w1th the vessel wall,

however, m1ght result 1n s1gn1f1cantly higher pressure fluctuat10ns,

accounting perhaps for the observat10n that beh1nd an arter1al

stenosis the vessel wall 1S often found to be weakened.

SIGNIFICANCE AND FUTURE STUDY. We have found from our

exper1mental measurements that vortex r1ngs behave in a viscous,

turbulent, or 1nvisc1d manner, depend1ng 1n large part on the Reynolds

number. Although other researchers have observed the V1SCOUS and

turbulent reg1ons, the existence of a group of vortex rings w1th an

1nvisc1d character had not been prev10usly estab11shed. L1kew1se,

although other researchers have observed single per10ds of breakdown

and rap1d change 1n veloc1ty for turbulent vortex rings, 1t is

s1gn1f1cant that trans1t10n per10ds have now been observed for vortex

rings at all Reynolds numbers, and 1n many cases more than one.

Such observat1ons have relevance not only wlth respect to the

evolutlon of slngle vortex rlngs, but wlth more compllcated flows

involvIng concentrated reg10ns of vort1c1ty. The dlfferent rates at

wh1ch circulat10n 1S lost from the core m1ght have lmplicat10ns w1th

respect to the evolut1on of more general vortex flows. The rapid

changes 1n vortex strength and the poss1ble 1nstabi11ty mechan1sms

assoc1ated w1th them mlght playa role in the production of smaller

scales of turbulence from larger scales, espec1ally for shear flows.

If found to be the case in more general turbulent flows, sudden


losses of vort~city as vortex tubes become unstable, break, and

reform, when averaged over long periods of time might produce large

contributions to a turbulent viscosity. In addition, since the sound

field in turbulent jets arises from non-steady oscillations in the

potential funct~on, sudden changes in vortex ring strength (and

therefore in the potential field) might be expected to result in the

production of noise. It would be of considerable practical interest,

therefore, to examine these periods of rapid change in vortex ring

strength in more detail, to determine what mechanisms are involved,

and to search for such mechanisms in the development of more general

turbulent flows.

Using the kinematic nature of vorticity we have developed a

method to calculate the total circulation of vortex rings from flow

visualization measurements. This technique is considerably simpler to

apply than previous methods which require knowledge of the velocity

field w~thin the vortex ring, and we now have the ability to compute

the time rate of change in strength for single vortex rings as they

propagate. With respect to the evolution of vortex rings, then, it

would be of interest to apply this technique to more extended studies.

For example, by vary~ng systematically the piston character~stics,

volume of fluid ejected, type of orifice used, etc. additional

relationships between the initial conditions and the loss of

circulation from the vortex r~ng as it evolves can be established.

We might note in closing that the technique we have developed to

calculate the total circulation of vortex rings from flow

visual~zat~on measurements can be applied Just as well to the study of

vortex pairs, with applicat~on, for example, to studies of tra~ling

vort~ces behind airfoils with f~n~te span.

APPENDIX A

VORTEX RING TRAJECTORY, SIZE, AND SHAPE PLOTS

for Runs 1 - 26

definitions:

x D1stance from the orifice. [cm]. r

t T1me. [sec.]

R Radius of vortex ring to core centerline.

R' Radius of vortex ring to outermost dyed region.

T Thickness of vortex r1ng to outermost dyed region.

V Volume of vortex ring, 4/3 7T R ,2T

Vej Volume of fluid eJected through the orifice.

Table A_1: Summary of exper1mental values for each run.

Table A-2: Summary of regime slopes for each run.

F1gures A-1 to A-26:

a) Vortex r1ng trajectory plots

b) Vortex r1ng size and shape plots

135

Appendix A 136

d (sec. ) (in. ) (em/sec) (em) ReT T X u Re

j uj Run D P P p. d

1 dye .1000 5598 .169 .300 4.509 55978 450.9 1.188

2 dye .1000 4254 .172 .232 3.426 42540 345.6 1.188

3 dye .1000 2471 .208 .163 1.990 24712 199.0 1.188

4 dye .1497 5840 .162 .300 4.704 38931 209.05 1. 778

5 H2 .1497 5468 .173 .300 4.405 36456 195.8 1. 778 -

6 dye .1497 4426 .166 .233 3.565 29508 158.4 1. 778

7 H2 .1497 3858 .188 .230 3.107 25719 138.1 1. 778

8 H2 .1497 3719 .195 .230 2.996 24796 133.2 1. 778

9 dye .1497 2650 .194 .163 2.134 17664 94.8 1. 778

10 H2 .1497 2675 .191 .162 2.154 17831 95. 75 1. 778

11 dye .2525 6224 .152 .300 5.013 24649 78.63 3.000

12 H2 .2525 5419 .174 .299 4.365 21461 68.38 3.000

13 dye .2524 2623 .196 .163 2.112 10388 33.13 3.000

14 dye .3207 5406 .175 .300 4.354 16857 42.33 3.810

15 dye .3207 4025 .178 .230 3.282 12550 31.91 3.810

16 dye .3207 2647 .193 .162 2.132 8254 20.73 3.810

17 dye .4209 6143 .154 .300 4.948 14596 27.94 5.000

18 H2 .4209 5409 .172 .295 4.356 12852 24.62 5.000

19 dye .4209 4266 .170 .230 3.436 10136 19.40 5.000

20 dye .4811 5468 .174 .300 4.405 11366 19.04 5.715

21 H2 .4811 5715 .165 .299 4.603 11881 19.88 5.715

22 dye .4811 3871 .189 .232 3.118 8048 13.48 5.715

23 dye .4811 2554 .200 .162 2.057 5310 8.891 5.715

24 dye .6414 5005 .189 .300 4.032 7803 9.800 7.620

25 dye .6414 3899 .186 .230 3.141 6079 7.635 7.620

26 dye "'.3 "'13470 "'.070 .299 10.85 "'44900 "'120 Ad/~ "'.92

TABLE A-I

Summary of experimental values for each run

Run Al A2 A3 Bl B2 B3 Y1 Y2 Y3

1 23.244 27.636 2 22.803 28.364 3 24.236 26.948 (70.128) 4 17.253 23.240 (28.725) 85.741 5 15.997 18.721 (rocking oscillation) 6 110.20 88.346 7 101.52 86.500 8 turbulence 9 83.686 70.421 63.520 } repeatability 10 83.836 70.435 62.410

11 41. 363 37.861 12 35.807 32.448 28.995 13 19.249 17.016 14 18.145 16.676 14.358 15 14.532 13.715 13.056 16 7.175 6.518 6.017 17 9.813 8.866 8.550 18 8.417 7.816 19 6.694 5.944 20 4.819 4.396 21 6.252 22 3.853 (3.170) .8280 23 (1. 562) .8239 .7650 24 .8664 .7313 25 .8623 .7681 26 3.913 5.974 10.873 (irregular occlusion)

-- -- ---- ------------

TABLE A-2 group 1:

Summary of regime slopes for each run group 2:

group 3:

(x) = em

C1 C2 C3

1.568 2.529 2.875 2.202 2.989 2.093 2.589

x = A In t + const. n x .. B t + const. n In x - Y In t + In C n n

(U) = em/sec.

A U - ...!! t U - B n y-1 U-YCt n

n n

n - regime

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N

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Appendix A 138

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Appendix A 139

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Appendix A 140

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Re. = 38931 J

FIGURE A-Sa Run 5 dID = 0.15 Rocking oscillation. 1I2

(WO)JX

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= 36456 (Section 4-4.3)

Appendix A 146

Appendix A 147

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Appendix A 148

Appendix A 149

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Appendix A 150

FIGURE A-7a Run 7 dID = 0.15 Re. 25719

J H2

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Appendix A 151

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The effect of turbulence in the ambient fluid. Upper points: Run 7 Lower points: Run 8

Appendix A 152

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Appendix A 153

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Appendix A 156

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Appendix A 157

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Appendix A 159

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Appendix A 160

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(WO).lX

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Appendix A 161

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APpendix A 162

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Appendix A 166

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Appendix A 167

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Appendix A 169

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Appendix A 170

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Appendix A 171

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Appendix A 172

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Appendix A 173

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co 0

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FIGURE A-20b Run 20 diD = 0.48 Re. = 11366

J Dye

Appendix A 176

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n;'TTlr o.

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Appendix A 177

o

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Appendix A 178

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Appendix A 181

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(WO)..IX uI

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Viscous interaction with the tube wall, and reformation (Section 4-4.3).

Appendix A 182

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§ ... c ....

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r

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(I)

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Appendix A 183

Appendix A 184

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0.64 Re. J

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Appendix A 185

o

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-.. o o

., I

t I

col I

f I r I p I .. I

r I -~ o Q)

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,~/l

The "apparent" decrease in R' and R is due to the bending back of the vortex ring (Section 4-2).

FIGURE A-26 Run 26 Ad/~- 0.92 Dye

Ret' 44900

Irregular occlusion (Section 4-4.3 and 5-4).

Appendix A 186

o II) 10 ...... II)

E

c ....

APPENDIX B

VORTEX RING STRENGTH AND VELOCITY PLOTS

Table B-1: Initial vortex ring parameters for each run

Figures B-1 to B-17: Computed values of total circulation as a

function of time plotted with the vortex ring

velocity. Total circulation, or strength: solid curves Vortex ring velocity: dashed curves

187

Appendic B 188

(sec. ) 2 (em Isec.)

Run t

0 £ ro Re

R Re r

1 .112 .19 827 50144 89560

2 .102 .13 731 35091 76393

3 .170 .18 684 31050 71481

4 .122 .20 756 35233 79005

5 .145 .28 38045

6 .100 .25 584 33858 61030

7 .126 .26 33101

8

9 .126 .23 460 23421 48072

10 .136 .25 26459

11 .186 .30 311 15181 32501

12 .180 .31 13845

13 .370 .31 118 7455 12331

14 .450 .42 220 9530 22991

15 .500 .40 120 7217 12540

16 .960 .36 52.2 3197 5455

17 .583 .38 100 4641 10450

18

19 1.327 .47 64.6 3894 6751

20 1.489 .38 54.2 2291 5664

21

22 1.170 .45 34.5 2205 3605

23 3.00 .44 14.5 937 1515

24

25 3.693 .37 13.0 688 1359

26

TABLE B-1

Initial vortex ring parameters for each run

C :::J L

o <0 N

o N N

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Appendix B 189

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..

N

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Appendix B 190

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(l)

E

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(V')

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Appendix B 192

o <.0

a ('oJ (V)

IJ') (\")

• o

o (\")

• o

()

L() (l) (\J cJ) · ........... o

o (\J

• o

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(l)

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C :J L

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(oeSjWO) Al1Q01eA o o

Appendix B 193

o 11)

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FIGURE B-S Run 6

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Appendix B 194

o o N

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• o Q)

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0 0 0 oo::r oo::r (V)

0 0 CO N N N

Appendix B 195

o N

-t

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---

0 ('

• 0

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.,........,

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cJ) • 0 ............

(1)

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(V) ....-

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r/ · -I:;;"~.J I.' • I ~ I

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Appendix B 196

o

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•

0 (V)

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.....--0 (J) cJ)

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(J)

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g-+-> •

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FIGURE B-8 Run 13

a (0

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FIGURE B-9 Run 14

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j If I :' I:

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Appendix B 197

00

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Appendix B 198

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o L()

• o

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~~~-4 __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~~ __ ~O

o ({)

FIGURE B-lO Run 15

o 00

o ({)

Appendix B 199

(09S/WO) '\t 1OO 19 /\ <0 N 00 ~ 0 •

" II II 0 II 0

• ,I ~ ,I I I I 0 I L() I • I M

:1: ~-,.' ~!;:' 0

" . 0 " . • I' M • ~

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(08S/ zWO) Lft 6u8 .J1 S

FIGURE B-ll Run 16

(' ~

C :J L

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FIGURE B-12 Run 17

(09S/WO) ":}1 00 19 /\ (J) ...- (V) d • • ~ m ('.

I

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Appendix B 200

L()

• L()

0 L()

• (V)

0 0 •

(V)

0 L()

• N ---..

0 (J)

0 if) 0 · -...-N

(J)

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~+-> ...-

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FIGURE B-13 Run 19

00 •

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Appendix B 201

o o

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(J)

(l)

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O+-> • (\J

o o • ..-

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C :J L

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(09S/~WO) "

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Appendix B 202

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Appendix B 203

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• • • • .0

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Appendix B 204

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Q)

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FIGURE B-17

Run 25

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I I I I I I I I I

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Appendix B 205

r-.. • o

o • o

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Bibliography

1. Arms, R.J., Hama, F.R. (1965), "Loca11zed-Induction Concept on a Curved Vortex and Motion of an Elliptic Vortex Ring," ~ Fluids , .§. , 553.

2. Batchelor, G.K. (1967), An Introduction.t..Q Fluid Dynam1cs , Cambr1dge Un1vers1ty Press.

3. Brasseur, J. G., Chang, I-D. (1979), "Flow Visua11zation of the Post-Stenot1c Flowfield in Pulsatile Flow," Proc . .19li B1omech. ~ , 157.

4. Brasseur, J.G., Koutsoyann1s, S.P., Chang, I-D. (1976), "Formation and Propagat10n of Vortex Rings in AX1symmetr1c, Bounded, Pulsed Flow," SUDAAR Report No. 502, Stanford University.

5. Cassanova, R. (1975), "An Experimental Invest1gation of Steady and Pulsatile Flow Through Partial Occlusions in a Rigid Tube," Ph.D. Dissertation, Georgia Inst. of Technology.

6. Chang, I.C., Chang I-D. (1968), "Potential of a Charged Sphere Inside a Grounded Cyl1ndrical Tube, II .J...... Math . .arul Physics , .!IT. , 366.

7. Chen, C.-J., Chang, L.-M. (1972), "Flow Patterns of a Circular Vortex R1ng with Density D1fference Under Gravity," .J...... .Alml... Mechanics, December, 869.

8. Cone Jr., C.D. (1962), "The Soaring Flight of B1rds," Scient1fic American, April, 659.

9. Davies, P.O.A.L., Yule, A.J. (1975), "Coherent Structures 1n Turbulence," .J...... .E..l.!il.!l Mech. , .6.9.. , 513.

10. Dyson, F.W. (1893), "The Potenhal of an Anchor Ring," Trans . .R.QL,. SQQ.... London , ~ , 1041.

1'. Feynman, R.P. (1957), "Applicahon of Quantum MechaniCS," Progress in ~ Temperature Physics, Vol. 1, Ed.: C.J. Gortner, North-Holland Pub. Co., Amsterdam, p.17.

12. Fineman, J.C., Chase, C.E. (1963), "EnergyofaVortexRing1na

206

Bibliography 207

Tube and Crit1.cal Veloc1.ties 1.n L1.qu1.d Hel1.urn II," ~ ~ , J29.. , 1.

13. Fohl, T. (1967), "Turbulent Effects 1.n the Format1.on of Buoyant Vortex Rings," J........Al2l2l.... PhYS1CS ,3§. ,4097.

14. Fohl, T. Turner, J.S. (1975), "Colilding Vortex Rings," Phys. Fluids , ~ , 433.

15. Fraenkel, L.E. (1970), "On Steady Vortex Rlngs of Small Cross-Section 1.n an Ideal Flu1.d," Proc. li2Y..... ~ London , .A3..2..Q. , 29.

16. Fraenkel, L.E. (19 rr2), "Examples of Steady Vortex Rings of Small Cross-Sectl.on l.n an Ideal Fluid," J........E.JJ.li.g Mech. ,.5.l , 119.

17. Frydenlund, O. (1974), "An Experimental Investigatl.on of Heavy Vortex Rings Rl.sing Against Gravity," Physlca Norvegl.ca , I , 113.

18. Gopal, E.S.R. (1963), "Motion and Stabl.lity of Vortl.ces in a Fl.llite Channel: Appll.cation to Ll.quid Hell.urn II," .Ann..... ~ , ..25.. , 196.

19. Hecht, A.M., Bilanin, A.J., Hl.rsh, J.E., Snedeker, R.S. (1979), "Turbulent Vortices in Stratl.fied Flul.ds," AIAA Paper '79-0 151 .

20. Helmholtz, H. (1867), "On Integrals of the Hydrodynaml.cal Equations whl.ch Express Vortex Motion," Phil . .Ma&.... , 33. , 485.

21. Hl.cks, W.M. (1884), "On the Steady Motl.on and Small Vibrations of a Hollow Vortex," Phl.l. Trans . .fi.Qy..SQQ.... London,.A!15. , 161.

22. Hicks, W.M. (1885), "Researches on the Theory of Vortex Rings," Phl.l. Trans . .fi.Qy..~London,.A..11Q., 725.

23. Hl.cks, W.M. (1922), "On the Mutual Threadl.ng of Vortex Rl.ngs," Proc • .fi.Qy.. ~ London , ~ , 111.

24. Hill, M.J.M. (1894), "On a Spherical Vortex,".f.h.ll...... Trans . .fi.Qy.. ~ London , ~ , 213.

25. Johnson, G.M. (1971), "An Empirl.cal Model of the Motlon of Turbulent Vortex Rings," .AIAA. Journal , .9. , 763.

26. Kambe, T., Takao, T. (1971), "Motion of Dl.storted Vortex Rings," ~ Phys. ~ Japan, 3...!. , 591.

27. Kambe, T., Oshima, Y. (1975), "Generatl.on and Decay of Viscous

Bibliography 208

Vortex Rings," J....... .E..hys....... SQQ......sLal2.rul , .3li. , 271.

28. Krutzsch, C.-H. (1939), "Uber eine experimentell beobachtete Erscheinung an Wirbelringen bei ihrer translatorischen Bewegung in wirklichen Flussigkeiten," ~.E..hys....... , .35.. , 497.

29. Lamb, H. (1932), Hydrodynamics, Dover.

30. Leiss, C., D1dden, N. (1976), "Exper1mente zum Einfluss del" Anfangsbedingungen auf die Instabil1tiit von Ringwirbeln," k.. angew. Math. ~ , 5Q , T206.

31. Linden, P.F. ('973), "The Interaction of a Vortex Ring with a Sharp Density Interface: a Model for Turbulent Entra1nment," J....... Flu1d Mech. , ..6..Q. , 46'7.

32. Magarvey, R.H., MacLatchy, C.S. (1964), "The Format1on and Structure of Vortex R1ngs," ..c..an..... .sL...E..hys....... , ~ , 678.

33. Magarvey, R.H., MacLatchy, C.S. ('964), "The Dis1ntegrat10n of Vortex Rings," ~ J....... ~ , 1!2. , 684.

34. Maxworthy, T. (1972), "The Structure and Stabi11ty of Vortex Rings," .sL.. Flu1d Mech. ,.51. , '5.

35. Maxworthy, T. (1974), "Turbulent Vortex Rings," .sL.. ~ Mech. , ~ , 227.

36. Maxworthy, T. (1977), "Some Experimental Stud1es of Vortex Rings," .sL.. .E.ll.!.iQ. Me ch . ,..at , 465.

37. Meng, J.C.S. (1978), "The PhYS1CS of Vortex Ring Evoluhon 1n a Stratified and Shearing Environment," .sL.. Fluid Mech. , 84 , 455.

38. Miloh, T., Schlein, D.J. ('977), "Passage of a Vortex Ring Through a C1rcular Aperture in an Infin1te Plane," ~ Fluids, .2Q. , 1219.

39. Mohring, W. (1978), "On Vortex Sound at Low Mach Number," J........E.JJ.lll Mech. ,.§2. , 685.

40. Moore, D.W. Saffman, P.G. (1974), "A Note on the Stability of a Vortex Ring of Small Cross Section," Proc . .R.QL. SQQ..... London , .A3.31i , 535.

41. Morton, B.R. (1960), "Weak Thermal Vortex Rings," .sL...E.lu1.Q Mech. , .9. , 107.

Bibliography 209

42. Norbury, J. (1972), "A Steady Vortex Ring Close to H~ll's Spher~cal Vortex," Proc. ~.f..hJ.l....SQQ..... , 12. , 253.

43. Norbury, J. (1973), "A Fam~ly of Steady Vortex R~ngs,".J....... Flu~d Mech. ,51... , 417.

44. O'Brien, V. (1961), "Steady Sphero~dal Vort~ces--More Exact Solutions to the Navier-Stokes Equat~ons," Quart. AlmL.. Math. , 1.9. , 163.

45. Oka, S. (1936), "On the Instabil~ty and Break~ng Up of a R~ng of L~qu~d ~nto Small Drops," Phys~co-Math . .SQQ..... Japan , .!.a , 525.

46. Okabe, J., Inoue, S. (196 1), "The Generat~on of Vortex Rings," ~ ~ Inst. ~ Mech. , Kyushu Univers~ty, ~ , 91 •

47. Okabe, J., Inoue, S. (196 1), "The Generat~on of Vortex R~ngs II," ~~ Inst. Appl. Mech. , Kyushu Un~vers~ty, ~ , 147.

48. Osh~ma, Y. (1972), "Mot~on of Vortex Rings ~n Water," .J....... ~ ~.sI..rumn , .3Z. , 1125.

49. Osh~ma, Y., Kambe, T., Asaka, S. (1975), "Interact~on of Two Vortex R~ngs Mov~ng Along a Common Axis of Symmetry," .J....... ~ ~ Japan, 3a , 1159.

50. Phill~ps, O.M. (1956), "The Final Period of Decay of Non-homogeneous Turbulence," ~ ~ .f..hJ.l... ~ , ..5.2.. , 135.

51. Pullin, D.I. (1979), "Vortex R~ng Formation at Tube and Orif~ce Openings," Phys. FlUl.ds , .2..2. , 401.

52. Rayner, J.M.V. (1979), "A Vortex Theory of An~mal Flight, Part 1 and Part 2," .J....... Fluid Mech. ,.9.1 , 697 and 731.

53. Riley, N. (1974), "Flows with Concentrated Vorhcity: A Report on EUROMECH 41 ," .J....... FlUl.d Mech. , .Q£ , 33.

54. Saffman, P.G. (1970), "The Velocity of VISCOUS Vortex Rings," Stud. Appl. Math. , ~ , 37 1 •

55. Saffman, P.G. (1975), "On the Formation of Vortex Rings,".s.t.u.d.... Appl. Math. , luI 54 , 261.

56. Saffman, P. G. (1978), "The Number of Waves on Unstable Vortex Rings," .J....... Flu~d Mech. , ~ , 625.

57. Sallet, D.W., W~dmayer, R.S. (1974), "An ExperImental Investigation of Laminar and Turbulent Vortex Rings in A~r," z... Flugwiss ,2a , 207.

Bibliography 210

58. Sa11et, D. W. (1975), "Impulsive Motion of a C~rcular Disk Which Causes a Vortex Ring," ~ Fluids , ..!Ji , 109.

59. Shlien, D.J., Thompson, D.W. (1975), "Some Experiments on the Mot ion of an Isolated Laminar Thermal," "L. .E..lY.id. Mech. ,12.. , 35.

60. Sull~van, J.P., W~dnall, S.E., Ezekiel, S. (1973), " Study of Vortex Rings using a Laser Doppler Velocimeter," AIM Journal, 11. , 1384.

61. Taylor, G.I. (1953), "Formation of a Vortex Ring by Giving an Impulse to a Circular Disk and Then Dissolving it Away," .sL.. ~flWh, ~, 104.

62. Thomson, J. J. (1883), A Treatise .QIl .tM Motion ~ vortex .B..in&:l , MacMillan and Co., London.

63. Thomson, J.J. (1885), "On the Formation of Vortex Rings by Drops Falling into Liqu~ds and some Allied Phenomenon," Proc. Roy.Soc. London , ~ , 41 7.

64. Thomson, W. or Lord Kelvin (1910), Mathematical ~ Phys~cal Papers , Vol. IV, Art~cles No.1, 2, 3, 10, Cambridge University Press.

65. Tob~n, R., Chang, I-D. (1976), "Wall Pressure Spectra Scaling Downstream of Stenoses ~n Steady Tube Flow," .sL.. B~omech. ,.9. , 633.

66. Tsai, C-Y., Widnall, S.E. (1976), "The Instability of Short Waves on a Straight Vortex Filament in a Weak Externally Imposed Strain Field," .sL.. .E:.llUJl Mech. , 1.1 , 721.

67. Tung, C., Ting, L. (1967), "Motion and Decay of a Vortex Ring," ~ Fluids, 1.Q. , 901.

68. Turner, J.S. (1957), "Buoyant Vortex Rings," Proc . .R.2L. SQ.Q...... London , ~ ,61.

69. Turner, J.S. (1960), "A Comparison Between Buoyant Vortex R~ngs and Vortex Pairs," .sL.. ~ Mech. ,1. , 419.

70. Turner, J. S. (1964), "The Flow ~nto an Expand~ng Shperical Vortex," .sL.. Fluid Mech. ,1.li , 195.

71. Van Dyke, M. (1974), "Analysis and Improvement of Perturbation Series," ~ Mech. App!. Math. ,2.l. , 423.

Bibliography 211

72. Walraven, A. (1970), "Energy and Motl.on of Vortex RJ.ngs in Ll.quid Hell.um II l.n the Presence of Varl.ous Plane Obstacles," ~ .furr... , 1. , 145.

73. Wells, D.R. (1962), "Observation of Plasma Vortex Rings," £hY.:L. Fluids ,.5. , 1016.

74. Wl.dnall, S.E., Sullivan, J.P. (1973), "On the Stability of Vortex Rings," .fl::Q..Q.... li2Y.... SmL.. London , .A3.3.2.. , 335.

75. Widnall, S.E., Bll.ss, D.B., Tsal., C.-Y. (1974), "The Instabl.ll.ty of Short Waves on a Vortex Rl.ng," ~ .E.l.u.J.s! Mech. ,QQ. , 35.

76. Widnall, S.E. (1975), "The Structure and Dynamics of Vortex Filaments," .Ann..... ~.E.l..... ~ , ~ , 141.

77. Wl.dnall, S.E., Tsal., C.-Y. (1977), "The Instability of the Thin Vortex Ring of Constant Vorticl.ty," Phil. Trans. li.2Y....-~ London , ~ , No. 1344.

78. Winant, C.D., Browand, F.K. (1974), "Vortex Pairl.ng: The Mechanl.sm of Turbulent Ml.xing-layer Growth at Moderate Reynolds Number," ~ FlUl.d Mech. ,.B , 237.

79. Yamada, H., Matsu, T. (1978), "Preliminary Study of Mutual SlJ.p-Through of a Pair of VortJ.cl.es," ~ Flul.ds , .£1. , 292.

The true artist is a tiaeuLto

he paints with red-and-black ink

with black water

the true artist is wise

god is in his heart he paints god

into things

he knows all colors

he makes shapes he draws feet

and faces

he paints shadows

he is a Toltec

he has a dialogue with his own heart

-poems from ancient Mexico retold from Nahuatl texts

To Agl1e,6

End of Document

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