scholarly purposes may be granted by the Head of my Department or by his or her .... S p a t i a l i n t e r p o l a t e d u> and V profile from streaks shown in Fig.
AUTOMATED TWO DIMENSIONAL FLOW V I S U A L I Z A T I O N AND COHERENT STRUCTURE RECOGNITION by A L E X I S KAI-HON LAU B.Sc.
Chinese U n i v e r s i t y
o f Hong K o n g , 1984
A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s
We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the required
standard
THE UNIVERSITY OF B R I T I S H COLUMBIA July
©
1986
A l e x i s K a i hon
L a u , 1986
In
presenting
requirements
this for
thesis
an
advanced
British
Columbia, I agree
freely
available
that
permission
scholarly or
by
for
for
partial degree
that
the
reference
extensive
at
Library
and s t u d y .
copying
his
or
her
representatives. of t h i s
thesis
be a l l o w e d w i t h o u t my w r i t t e n
Department of
Physics
The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5
Date:
fulfilment
of
July
31
1986
of
shall
make
I further this
thesis
for financial
of i t
agree for
Department
I t i s understood
permission.
the
the U n i v e r s i t y
p u r p o s e s may be g r a n t e d by t h e Head o f my
copying or p u b l i c a t i o n not
in
gain
that shall
i i
Abstract T h i s p a p e r d e s c r i b e s an e f f i c i e n t quantitative It
also
data
introduces
identification fields.
of
a
The
stage
surface 10")
sequences of f l u i d
computer
coherent
automated
structures
flow
images.
method
in
f o r the
turbulent
of c o h e r e n t
structure recognition.
motion
turbulent
was
on
a
video
digitized
as
a
binary
t r a c k i n g and c o n n e c t i n g using
on
visualized
particles
frames
for extracting
flow
T h i s method e l i m i n a t e s s u b j e c t i v e m a n u a l j u d g e m e n t i n
the c r u c i a l
number
from time
method
a
by
tape.
recording Each
image
video
using
the tracer
mainframe
grid
a
images of t r a c e r frame
was
through
the
By
successive history
was
Streak
fitted
by
polynomials
g i v e v a r i o u s f l o w p a r a m e t e r s of i n t e r e s t
over
desired
flow times.
velocities
were
In
particular
were
flow
then
reconstructed. to
trajectories
(Reynolds
microcomputer.
paths
computer
flow
the
then
linear
and
d e t e r m i n e d a s s c a t t e r p o i n t s f r o m w h i c h mesh
f i e l d s were i n t e r p o l a t e d .
C o h e r e n t s t r u c t u r e s were
by
of
t h r e s h o l d i n g the f i e l d
interpolated structure average The
mesh
was linear
f i e l d s of l i n e a r
parametrized and
angular
angular
with
structure properties.
velocity.
velocities, the
each
and energy
then
identified Using
properties
velocities,
f l o w d y n a m i c s and i n t e r a c t i o n s a r e
these
angular
the
coherent of
size,
content.
discussed
using
The
system
recognition
for
energetics
of
was a
new
structure
initial
stage of g r i d
results
structures
were
predictions. present
two
in extending
studies.
using
dimensional the system
enhance based
on
data the
I t i s a l s o intended as a flow v i s u a l i z a t i o n
Applying
the
system
successfully
s t r u c t u r e s manually
these
compared
to
turbulence
turbulence, i t
Limitation
presented.
of
structures.
80% o f a l l t h e c o h e r e n t
Parameter
primarily
t o be u s e d f o r o t h e r
coherent
also
model
coherent
general technique
over
developed
automatically
to
and the
recognized identified. identified
w i t h e s t a b l i s h e d r e s u l t s and m o d e l and
system
possible
improvement
i s discussed.
Various
on
the
aspects
t o a three dimensional environment are
iv
Table of Contents Abstract L i s t of F i g u r e s Acknowledgement Chapter I INTRODUCTION 1.1 T r a d i t i o n a l A p p r o a c h 1.2 C o h e r e n t S t r u c t u r e s a n d F l o w V i s u a l i z a t i o n 1.3 P r o s p e c t i v e f o r Computer P r o c e s s i n g 1 . 4 Objective 1.5 C h a p t e r O u t l i n e s Chapter I I COHERENT STRUCTURES - THE D E F I N I T I O N 2.1 C o h e r e n t S t r u c t u r e M o d e l i n T u r b u l e n c e 2.2 D e f i n i t i o n b a s e d on C o h e r e n t V o r t i c i t y 2.3 V o r t i c i t y a n d A n g u l a r V e l o c i t y 2.4 The D e f i n i t i o n Chapter I I I THE EXPERIMENTAL SYSTEM 3.1 The T o w i n g Tank 3.2 R e c o r d i n g 3.3 D i g i t i z a t i o n 3.4 D a t a S t o r a g e and T r a n s f e r Chapter IV SYSTEM P R I N C I P L E S 4.1 N o i s e R e d u c t i o n a n d T r a c e r Image C e n t e r i n g 4.2 S t r e a k T r a c k i n g 4.3 G r i d I n t e r p o l a t i o n And C o h e r e n t S t r u c t u r e Recognition 4.3.1 Time W i n d o w i n g 4.3.2 P a r a m e t e r I n t e r p o l a t i o n 4.3.3 P r e l i m i n a r y G r i d I n t e r p o l a t i o n 4.3.4 P r e l i m i n a r y C o h e r e n t S t r u c t u r e R e c o g n i t i o n 4.3.5 D a t a R e f i n e m e n t 4.3.6 F i n a l G r i d a n d C o h e r e n t S t r u c t u r e s 4.4 S t r u c t u r e P a r a m e t r i z a t i o n Chapter V SYSTEM ALGORITHM 5.1 N o i s e R e d u c t i o n a n d T r a c e r Image C e n t e r i n g 5.2 S t r e a k T r a c k i n g 5.3 G r i d I n t e r p o l a t i o n a n d C o h e r e n t S t r u c t u r e Recognition 5.3.1 Time W i n d o w i n g a n d P a r a m e t e r I n t e r p o l a t i o n 5.3.2 G r i d I n t e r p o l a t i o n
i i vi viii 1 1 3 6 8 9 12 12 15 16 ....18 21 21 24 25 29 31 32 39 42 44 45 50 50 52 61 64 67 67 72 77 77 80
V
5.3.3 C o h e r e n t S t r u c t u r e R e c o g n i t i o n 5.3.4 D a t a a n d S t r u c t u r e R e f i n e m e n t 5.4 S t r u c t u r e P a r a m e t r i z a t i o n
80 83 84
C h a p t e r VI HARDWARE CONSTRAINTS AND EXPERIMENTAL PARAMETERS 85 6.1 H a r d w a r e C o n s t r a i n t s a n d E x p e r i m e n t a l P a r a m e t e r s ..86 6.2 H a r d w a r e a n d C o n t r o l P a r a m e t e r s Used , 94 Chapter V I I RESULTS AND DISCUSSIONS 98 7.1 E x p e r i m e n t a l R e s u l t s on I n i t i a l G r i d T u r b u l e n c e ...98 7.1.1 C o h e r e n t S t r u c t u r e s a t P r o d u c t i o n 99 7.1.2 S p o n t a n e o u s E n e r g y D e c a y R a t e o f C o h e r e n t Structures ; 115 7.2 D i s c u s s i o n s a n d R e c o m m e n d a t i o n s 117 7.2.1 L i m i t a t i o n s o f t h e P r e s e n t S y s t e m 117 7.2.2 E x t e n s i o n t o a 3D S y s t e m 123 Chapter
VIII
CONCLUSION
130
BIBLIOGRAPHY
132
APPENDIX A - PARAMETER EXTRACTIONS FROM FITTED TRAJECTORY A. 1 S t a t i o n a r y M o d e l : Vcm i s Z e r o A.2 D r i f t i n g M o d e l : Vcm i s N o t Z e r o APPENDIX B - USING THE PACKAGE AT UBC
1 33 134 135 136
vi
List
1.
Flow p i c t u r e
Figures
showing c o h e r e n t s t r u c t u r e s
2. C o h e r e n t s t r u c t u r e s dimension 3. D i a g r a m m a t i c
of
outlined
5
by o u r d e f i n i t i o n
i n one
drawing of t h e system hardware
20 22
4. The t o w i n g t a n k
23
5. Numbered 8 - n e i g h b o u r s o f a p i x e l
32
6. U n p r o c e s s e d c o n s e c u t i v e d i g i t i z e d 7. S u p e r i m p o s e d streaks 8. 1D s t r u c t u r e s
images
p l o t of t r a c e r c e n t e r s and recognised
33 tracked
41
from w ( r ) p l o t
51
9. I d e a l i z e d a n d m o d i f i e d V ( r ) a n d w ( r ) p l o t s f o r coherent structures 10. T y p i c a l p l o t o f s t r e a k s structure.
tracked
55
i n a coherent
11. S p a t i a l i n t e r p o l a t e d w a n d V p l o t i n F i g . 10
from s t r e a k s
56 shown 57
12. S p a t i a l i n t e r p o l a t e d u> a n d V p r o f i l e f r o m s t r e a k s shown i n F i g . 10 w i t h CM p o i n t a s p o i n t o f z e r o veloc i t y 13. 1D s t r u c t u r e s small peaks
identified
14. P l o t o f f i n a l
recognized Coherent s t r u c t u r e s
15. S m o o t h i n g 16. C o h e r e n t
b e f o r e and a f t e r removal of
68
at production
17. P l o t o f c a l c u l a t e d r o t a t i o n a l e n e r g y v s t o t a l for recognized structures 18. S p a t i a l p l o t s o f w ( r ) a n d V ( r ) f o r d i f f e r e n t situations
. . 62 63
considerations structures
58
100 energy 108 109
vi i
19. S p a t i a l p l o t
of V ( r ) f o r d i f f e r e n t s i t u a t i o n s
20. L o g - l o g p l o t o f i n i t i a l the s t r u c t u r e r a d i u s R 21.
P l o t of i n i t i a l
110
2
decay r a t e A as a f u n c t i o n
d e c a y r a t e A v s 1/R
2
of
113 114
viii
Acknowledgement I want t o t h a n k V i n c e n t B a r e a u f o r i n t e r f a c i n g t h e VCR and t h e m i c r o c o m p u t e r . The " d e s i g n e r s t a n d a r d " v i d e o r e c o r d e r support stand b u i l t by P a u l B u r r e l and M a c i e j K o w a l e w s k i g a v e me much freedom i n s e l e c t i n g the view a r e a w h i l e r e m a i n i n g r i g i d l y in place during t h e e x p e r i m e n t s . T h a n k s s h o u l d a l s o be g i v e n t o D i r k Townsend who h e l p e d t o c o r r e c t my E n g l i s h . A l Cheuck was always t h e r e t o get the s u p p l i e s i n the s h o r t e s t p o s s i b l e time and a l s o made s u r e t h a t t h e i n s t r u m e n t a t i o n was w o r k i n g a s d e s i g n e d . The p l a s m a g r o u p must be t h a n k e d f o r s u p p o r t i n g my work h e r e a n d a l s o f o r g i v i n g me t h e l a r g e amount o f c o m p u t e r and r e a l d o l l a r s u s e d t o d e v e l o p and t e s t run the programs. S p e c i a l t h a n k s must be g i v e n t o S t u a r t Loewen who e x p l a i n e d t o me t h e use o f a l l t h e e q u i p m e n t , s t a y e d and d i s c u s s e d w i t h me d u r i n g t h e e x p e r i m e n t s , and a l s o p r o o f - r e a d my d r a f t up t o t h e m o r n i n g he had t o l e a v e f o r h i s b r o t h e r ' s w e d d i n g . F i n a l l y , I have t o t h a n k my s u p e r v i s o r P r o f e s s o r Boye A h l b o r n who guided me t h r o u g h o u t t h e d e v e l o p m e n t of t h e s y s t e m and t h e w r i t i n g o f this thesis.
1
I. In
INTRODUCTION
s p i t e o f d e c a d e s o f r e s e a r c h , t u r b u l e n c e r e m a i n s one o f t h e
major
unresolved
standing flow and
problems
inability
method d e v e l o p e d 1.1 T r a d i t i o n a l
limits
researchers by R e y n o l d s
and
in
turbulent
t e c h n o l o g i c a l developments,
to
question
the
conventional
i n the 19th century.
predict
to d e f i n e a flow f i e l d
i t in detail
at a later
t h i s method i n v o l v e s t h e f o l l o w i n g 1) w r i t e down t h e N a v i e r - S t o k e s 3V/3t + and
Our l o n g -
Approach
T r a d i t i o n a l l y one t r i e s time
classical physics.
to give r e l i a b l e predictions
conditions c r i t i c a l l y h a s l e d many
in
(V-V)V
=
time.
at a given Basically,
steps:
equation
-V(P/p)
+
uV V 2
,
t h e e q u a t i o n of C o n t i n u i t y 3p/3t + V - ( p V )
2) Assume
=
0 .
incompressibility; p = c o n s t a n t and V-V
=
0 .
3) I n t r o d u c e t h e R e y n o l d s Re
and
reduce
=
pVL/u
4) S p l i t
,
the governing
3V/3t + the
number, Re
(V-V)V
velocity
equations =
-VP
and
+
to a dimensionless
(V V)/Re . 2
pressure
( z e r o e t h o r d e r ) and f l u c t u a t i n g
form
fields
(1storder)
into
the
components:
mean
2
V = V° + V _ o
p
taking original
, and
+ pi.
p
the
1
zeroeth
order
as
steady
state
s a t i s f y i n g the
equations: 3V°/9t + (V°-V)V° = - V P
0
+ (V V°)/Re ; 2
V-V° = 0 . 5)
Substitute
eliminating flow
them
most
back
into
zeroeth
the
order
1
V-V
terms t o get t h e f i r s t
+ (V -V)V° = - V P
1
1
except
the
perturbed
(V«V)V
system
boundary
term of
very
similar
becomes
V
equations
is
c o n d i t i o n s of i n t e r e s t .
and P
1
1
relation.
then
This There
relation
i s usually are
other
a s we
T h i s p r o b l e m h a s been w o r k e d
systems.
relations
have
However, t h e r e
is
first
two
+ (V -V)V°.
1
This
1
to
to
close
as
in
proposed little
the noted
variables effort
system
the
the
of
closure
approach,
and then
s p l i t the
is still
invoked.
for nearly
still
the
to
o f some f o r m
been
3
Then t h e m a j o r
referred
on
the
have
variations
but a c l o s u r e r e l a t i o n
particular
, and
1
subjected
one may e l i m i n a t e P f r o m t h e e q u a t i o n s
field,
2
H o w e v e r , i t s h o u l d be
b u t o n l y two e q u a t i o n s .
t o look f o r a general
equations.
+ (V V )/Re
1
to
(V°-V)V
t h a t t h e s y s t e m now becomes u n c l o s e d
e.g.
order
= 0 .
1
T h i s g i v e s two e q u a t i o n s
is
equations,
equation: 3 V / 3 t + (V°-V)V
V°,
original
a
century
f o r many progress
and
different i n the
3
search not
f o ra universal relation indeed
exist.)
Thus,
the u n i v e r s a l
turbulence
to
(which
may
that
any
many
research
must
occur
relation.
1.2 C o h e r e n t S t r u c t u r e s a n d F l o w During
t h e system
i t appears
fundamental breakthrough i n outside
to solve
the past
Visualization
twenty y e a r s ,
as a r e s u l t
o f t h e work o f
K l i n e e t a l ( l 9 6 7 ) , Crow a n d C h a m p a g n e ( 1 9 7 1 ) a n d many o t h e r s , a new p e r s p e c t i v e large-scale
seemed
vortex
to
be
motions
emerging. in
The
turbulent
discovery
shear
g e n e r a l l y a g r e e d t o be one o f t h e most i m p o r t a n t in the f i e l d on
f o r many y e a r s .
observations
flow. to
to
up
flows
that these motions
and t h e r e s e a r c h e r s '
how we c a n l e a r n more a b o u t
coherent
vortex
turbulence
motions
are
now
role
in
the
amount o f r e s e a r c h
acceptance
of
intrinsic
them.
collectively that
they
h a s been done a n d t h e r e
importance There
shear
work i s t o f i n d o u t
of
coherent
turbulent
flows.
definition
o f c o h e r e n t s t r u c t u r e s , o r on t h e i r
and
motions
Such
known a s play
t r a n s p o r t p r o p e r t i e s of the flow.
substantial
the
are
through
s t r u c t u r e s and s p e c u l a t i o n s a r e
important
vortex
i n n e a r l y every type of t u r b u l e n t
I t i s now r e c o g n i z e d
large-scale
developments
f i n d i n g s were b a s e d
i n t e r e s t as r e p o r t s of such
pile
turbulent
i s now
i n t u r b u l e n t boundary l a y e r flow but q u i c k l y
attracted general started
The o r i g i n a l
flows
of
i s , however, l i t t l e
significance to turbulent
flow.
an A
i s wide
structures
to
a g r e e m e n t on t h e exact
role
in
4
As
recently
difficulty
Nearly
visualization.
of
by
Hussain(1985),
method we u s e t o d i s c o v e r
a l l discoveries
i s to the
take
the
momentum.
advantage
of
The
major
enormous i n f o r m a t i o n
density.
a c l e a r anatomical which
i s not
the
structure"
so
evident
powerful
tracer
just
i n F i g . 1.
we
p r o f i l e and a p p r e c i a t e
the
energy
method l i e s
and
in i t s
i s t h e method o f v i s u a l Our
visual
analytic
t h a t no modern c o m p u t e r c a n idea
of
what
i s meant
by l o o k i n g a t a f l o w p i c t u r e
by like
C o h e r e n t s t r u c t u r e s on t h e t o p r i g h t
are o u t l i n e d manually. paths,
velocity,
Another
m a t c h i t . We c a n have a g e n e r a l
hand c o r n e r
time-exposure
d i s s e c t i o n of the system.
so
something
one shown
flow
picture
power
the
use
flow
experimentalist.
"coherent
or
A well conceived
a n a l y s i s u s e d by t h e is
studies
these
f l o w a n d t h e n m e a s u r e f r o m them t h e v a r i o u s typically
advantage
and
instantaneous
parameters of i n t e r e s t ,
presents
the basic
T h e r e a r e many v a r i a n t s i n t h e method b u t t h e
idea
pictures
out
stems from t h e v e r y
structures.
general
pointed
can
On c l o s e r
examination
observe the s t r u c t u r e ' s
the structure energetic
of
velocity
strengths.
5
-3 c m
I
1 - Flow p i c t u r e showing coherent s t r u c t u r e s .
Figure This
visual
faculties
like
irrelevant
data
inherit
in
rapidly
discover
input, s i m i l a r i t y surface
from our
(2D)
changing
we
p u r p o s e s , the information information
use
is is
of
surface
has
e n a b l e d us
environment. The
to
difficulty
is
work
the
correlation
This
us
that
to no
reliable
pictures for analytic
required
to
large. remaining and
of
survive
I t also allows
r e c o g n i t i o n i n a f a s t and
prohibitively obtained,
intricate
analysis, filtering
flow v i s u a l i z a t i o n
amount
many
correlation analysis.
ancestors
yet perform the
When
analysis,
and
incorporates
the c o h e r e n t s t r u c t u r e s .
m a c h i n e can way.
preception
visual
power we our
1
extract Even
problems
pattern
useful
when
this
of
data
recognition
are
6
still
difficult.
of t h i s
technique
difficulties
i n the e a r l y
in
i n f o r m a t i o n was I t was
H u s s a i n ( 1 9 8 3 ) p o i n t e d out
getting
t h e r e and
hard
data
(Ahlborn
handle
1902,
and
out.
that
developed
laser doppler
1922)
was
time,
economical
new
gave
It
techniques
is
c o r r e l a t i o n which i s v i t a l
t e c h n i q u e s , and of
structures.
very
by h o t w i r e clean,
suffered
hard
easy
significantly
the
visualization.
or to
easy
from
being
o n l y s i n g l e or
multi-
to
infer
a r r a y of s e n s o r s
to coherent
flow
century
from the
the
spatial
structure recognition.
limit
the
power of
s o , many r e s e a r c h e r s have t u r n e d b a c k
flow
There they
Recent developments i n t h i s
by C a n t w e l l and 1.3
fast,
the
of
E x p e r i m e n t a l i s t s took
i n f o r m a t i o n c o l l e c t e d by a l i n e a r
difficulties
The
away
the
replaced
in information yielding
data.
the
method.
method
in
gradually
data.
However, t h e s e
point velocity
method
the
the
early
anemometers which
restrictive
These
in
b e c a u s e of
h a r d d a t a c o u l d have been e x t r a c t e d .
During
visualization
too
2 0 t h c e n t u r y was
downfall
j u s t t h e e x c e s s i v e amount o f work t h a t d r o v e
experimentalists.
way
t h a t the
find
field
the
were
to
these the
coherent reviewed
Coles(1983).
P r o s p e c t i v e f o r Computer P r o c e s s i n g While
coherent
s t u d i e d , t h e most
structures
influential
revolution
was
taking
computers.
The
ability
were
being
identified
change
since
the
industrial
p l a c e - the development of h i g h of computers t o escape the
and
speed
limitations
7
o f human s p e e d and with
handle
unprecedented
c o m p l i c a t e d way,
has
c o r n e r s of s c i e n c e .
excessively" large
accuracy
in
a
amount
most
The
and
computational addition
research
to
theoretical
the
indispensible
has
emerged
traditional
its
extensively
introduction,
in fluid
been t o s o l v e
the
boundary
w i t h the hot
divison
the
Navier-Stokes conditions.
w i r e p r o b e s and
calculation
and
visualization
coherent of
work
to
tool
and
branch
experimental
computer
of
the
The
in and
Animated
data
still
researchers
with
g r a p h i c s i s used
the
anemometers),
crucial
storage, most
subjective
data a c q u i s i t i o n The
enormous
process
o f d a t a e x t r a c t e d and
reliability
of t h e q u a n t i t a t i v e r e s u l t s .
According
t o a r e c e n t computer l i t e r a t u r e
to
studies
Nevertheless,
the manual d a t a a c q u i s i t i o n accuracy
numerically
acquisition,
require in
used have
LDAs ( l a s e r d o p p l e r help
been
In e x p e r i m e n t a l
structures recognition stages. in
has
major e f f o r t s
equation
presentation.
experiments
t h e amount and and
research
of
dynamics s t u d i e s .
i s used e x t e n s i v e l y
judgement
advancement.
as a s e p a r a t e
model f l o w s as s i m u l a t e d e x p e r i m e n t s .
it
is easily
science.
Since
various
become an
deadlocked
tool
i s u s e d e x t e n s i v e l y f o r i t s own
c o m p u t e r has
or
f i n d the b e s t example w i t h i n
t h e c o m p u t e r c o m m u n i t y i t s e l f , as t h i s p o w e r f u l accessible
data
monotonous
p u s h e d f o r w a r d many o t h e r w i s e A c t u a l l y , we
of
amount
still
thus the
search
and
limits extent
into
the
8
INSPEC
(information
services
community), l i t t l e
work has
acquisition
flow
processes
in are
recognition the
still
i n flow
prejudice
of
identification
lack
results it
of a
the
for
most
structures in opinions approach
and
structure
image
processing
unpopularity
technique
flow
v i s u a l i z a t i o n and
needed t o
this thesis
involves
definition results
it
and
in
the
structures.
hard
to
compare
q u a n t i t a t i v e d a t a makes
may
of computer a n a l y s i s of
approach i s d e f i n i t e l y
This
about c o h e r e n t
a b s e n c e of e x t e n s i v e
data
coherent
it
makes
the
subjective
i n v e s t i g a t o r s both i n the the
engineering
Manual
hard to decide which approach i s b e t t e r .
the
an
visualization.
general
and
and
v i s u a l i z a t i o n experiments.
widespread d i f f e r e n c e s The
physics
been done i n a u t o m a t i n g
used
the of
f o r the
be flow
fully
The a
newness
reason
for
visualization. utilize
the
is a preliminary
of the Such
power
of
attempt
in
this direction. 1 .4 Object ive As the
w i l l be
i n more d e t a i l
immediate m o t i v a t i o n
visualization S.
described
study used
Loewen(1985).
turbulence coherent
coherent
by
They
b a s e d on
procedure structure
t h i s work was my
energy My of
a and
next
B.
statistical size interest
data
acquisition, After
was
flow
Ahlborn
and
model
for
spectrum
initial
recognition.
chapter,
t o enhance the
supervisor,
proposed
the
structures.
t h e i r manual
of
i n the
to
of
the
automate
analysis studying
and the
9
literature,
the
appreciated,
general
need
as one was a i m e d
while
other
c o m p r o m i s e was on
possible
was
that
the
F u r t h e r m o r e , as t h i s
to
be
intended whenever
energy
variations
be
specific
that
parameters, established,
was
general
m o d e l , i t w o u l d be c l e a r l y
could
be
isstill
environments.
by
and
made
comparing
sometimes efficient use.
considerations
in
The were
noted
other
how
systems.
a new m e t h o d , a n o t h e r a i m was t o of such systems i n t h e r e a l
The a c t u a l w o r k i n g d e s i g n s s h o u l d
i n t i m a t e l y r e l a t e d t o the experiments
hoped
system
specific
f o r the
evaluate i n d e t a i l the f e a s i b i l i t y experimental
a
T h e s e two r e q u i r e m e n t s were
conflicting
focused
such
a n d i t was d e c i d e d t o d e s i g n t h e s y s t e m t o s e r v e
as a g e n e r a l t e c h n i q u e .
the
for
the
some g e n e r a l g u i d a n c e
of i n t e r e s t .
It
was
importance
of
the
relative
for further
design
can
be
thereby i n c r e a s i n g the cost e f f e c t i v e n e s s of the
system. 1.5
Chapter O u t l i n e s In
the next c h a p t e r , the p r i n c i p a l
ideas i n
Loewen's work a r e g i v e n , f o c u s i n g on t h e i r structures. an
example
Then t h e d e f i n i t i o n of
one
of
the
structures i n the l i t e r a t u r e . this
and
concept of coherent
by Hussain(1983) i s g i v e n a s
current
concepts
of
F i n a l l y , the d e f i n i t i o n
work i s g i v e n , a n d i t s r e l a t i o n
and t o H u s s a i n ' s d e f i n i t i o n
Ahlborn
coherent used i n
to the s t a t i s t i c a l
i s discussed.
model
10
The
t h i r d chapter
system. rather
Emphasis
and
because
partly
hardware
the
of
the
on what must be done i n e a c h
step
a c t u a l designs
designs.
are very
contribution in
their
V
software
of t h e s y s t e m i s
separately.
Chapter regarding
implementation.
The
IV
considerations.
This
presents
development
was
i n the
next
next
chapter
to
do
j u s t confuse
are s e p a r a b l e .
the
reader
the
methods
understood
well
Chapter various affect
VI
are
giving
the v a r i o u s
to enable
hardware
to
parameters
setting. application
between the
They of
have the
to
the
together
Both
be
software
principles have t o
regarding
package.
be
the
c o n t r o l s that
This serves to
for
and
successfully.
experimental
matched
of
would
principles
separate.
s y s t e m h a r d w a r e and
that
algorithms
considerations and
of
efficiency
T h i s d o e s not mean t h e
t h e p e r f o r m a n c e of t h e s y s t e m .
relation
the
t h e s y s t e m t o be u s e d
i s devoted
with
done i n t h e b e l i e f
M i x i n g t h e two
completely
IV
constraints
i n understanding
the a l g o r i t h m fundamentals.
chapter
e l a b o r a t e s the a l g o r i t h m
s e p a r a t i o n was
it
in
the p r i n c i p l e s
t h e p r i n c i p l e s o f what s h o u l d be done and
the
is
system dependent
described
various
(not the programming t e c h n i q u e )
and
This
m a j o r work i s e l a b o r a t e d i n d e t a i l
considerations
how
part
chapters. The
and
The
put
the
the a c t u a l hardware
b e c a u s e my
minimal. three
is
than d e t a i l i n g
partly
describes
clarify
the
experimental
the
successful
Results
of
an
11
investigation grid
of coherent
turbulence
d i s c u s s i o n o f how
are given our
studies.
A
Information
a b o u t how
Appendix
B.
s t r u c t u r e i n the i n the seventh
system
conclusion
is
t o use
may then
be
initial chapter
extended
drawn
in
t h e p a c k a g e a t UBC
period
of
along with a for
further
Chapter is
VII.
given
in
12
II. The
initial
COHERENT STRUCTURES - THE D E F I N I T I O N motivation
of
this
work
was
t o automate t h e
t u r b u l e n c e s t u d y s t a r t e d by A h l b o r n a n d Loewen i n following connect 2.1
i s given
as
of t h e i r
r e c e n t paper t o
S t r u c t u r e Model i n Turbulence
T h i s model d i f f e r s the very s t a r t i n g
large-scale in
summary
The
t h i s work a n d t h e i r s .
Coherent
from
a
1983.
energy
from t r a d i t i o n a l point.
balance
I t i s based
r a t h e r than
the Navier-Stokes equation.
features
of
turbulent
distributions
of
coherent
structures
flow
from
To a v e r y are
r o t a t i n g columns of f l u i d
research
on c o n s i d e r a t i o n o f
l o c a l momentum b a l a n c e
I t attempts
t o describe "gross
statistically
interacting
( A h l b o r n e t a l 1985).
turbulence
coherent
crude
averaged structures"
approximation,
c o n s i d e r e d as i d e a l i z e d c h a r a c t e r i z e d by
the
these
cylindrical
radius R
and
t h e a n g u l a r v e l o c i t y u>. They i n t e r a c t fluid,
w i t h t h e s u r r o u n d i n g s i n t h e forms of eddy-
eddy-flow
and
eddy-eddy
interactions,
"eddy" u s e d a n a l o g o u s l y w i t h " c o h e r e n t of
transfer
of
energy
these
coherent
structures
the s i z e spectrum distributions
The
rate
s t r u c t u r e s due t o t h e
by A, B and C r e s p e c t i v e l y .
With
i n m i n d , t h e f l o w i s d e s c r i b e d by
N(R) o r t h e
can
structure".
of the coherent
above p r o c e s s e s a r e denoted
w i t h the term
energy
spectrum
be i n t e r p r e t e d a s p r o b a b i l i t y
N(E).
These
functions or
13
ensemble averages.
System i n t e r a c t i o n s are then
i n t o a s e t of r a t e e q u a t i o n s f o r the energy
incorporated
distribution
N(E^)
as: (2.1)
d(Ni.)/dt = I
where The
= N(E-^)
(A-.jN:) + E
and
t h e summation i s over
cylindrical
right
Reynolds bodies eddies
ideas
coherent
number of e s t a b l i s h e d f o r m and
function N
and
in
and
their
structures, theoretical
or the
crude
j and
k. from
interaction
onset
boundary
of
layers;
and
coefficient useful
layer
for bluff
model
flow
bodies.
2) t h e s i z e
in obtaining
Thus,
1
and
is
suffers
prohibitively from
smallest
proves
to
However, t h e a c t u a l
being
large. subjective
be
a
a and
amount
the
Moreover, and
of
4) t h e d r a g
required to obtain experimental data l i k e
distributions analysis
of the
to
bluff
distribution
a wall; this
behind
a
1) t h e
the g r o s s f e a t u r e s of the flow
more r i g o r o u s work i s w o r t h w h i l e . work
in deriving
These i n c l u d e
instability
near
of
experimental results
i n a f l o w ; 3) t h e l o g a r i t h m i c v e l o c i t y boundary
assumption
they succeed
o r d e r of m a g n i t u d e .
number f o r t h e
turbulent
of
indices
coefficients. With these major
the
(C^.N^N.)
g r o s s p r o p e r t i e s o f t h e f l o w a r e t o be d e r i v e d e i t h e r
t h e moments o f t h e d i s t r i b u t i o n rate
(B^-N-) + E
energy manual
person's
In 1983, S. Loewen spent about 2 months o f m a n u a l l a b o u r a n a l y z i n g o v e r 2000 s t r u c t u r e s w i t h a b o u t 30,000 individual s t r e a k s from time exposure p i c t u r e s .
14
consistency automated
suffers
from
o b j e c t i v e of
t o , we
a
this
computer can
have
to
give
structure should
he be
streamlines
the
instantaneous
a
This
clear
recognize
"assumed
the
i n the
making
it
analytic
(tracer
This
i s not
rotational the
these
and
on a n g u l a r first the
to
structures
them.
that
kinetic
of
for
match
process the
is
same i s s u e .
of
the
The
use
constant
ignores
not
the
i s that
readily
t h i s method i s f r a m e be
an
dependent
a Galilean variant.
"assumption
that
i s no
dependent
(Ahlborn
longer
the
et a l 1985).
l a t e r , another d e f i n i t i o n Before
the
clear,
s t r u c t u r e s so d e f i n e d w i t h
energy i n the eddies
reasons given
work
structure
remain
This
defined w i l l
v e l o c i t y u> i s u s e d .
t u r n our
coherent
coherent
Another disadvantage
Finally, so
for
periphery
interest
exposure.
f r a m e " i n t h e model
other
primary
In the p r e v i o u s
the
i n l i n e w i t h the u n d e r l y i n g
inertial
the
definition
paths)
recognition
definition.
the
on
time
hard
and
became
largest closed streamlines."
information.
physics
an
s t r u c t u r e and e x t r a c t i n g
r e c o g n i t i o n assumes t h e p a r a m e t e r s o f throughout
that
o n l y do e x a c t l y what i t i s programmed
it
i t can
Loewen(1983),
of
be d e s i g n e d .
appeared
work.
structures before by
It
system f o r r e c o g n i z i n g the
i t s parameters should
As
fatigue.
looking into
a t t e n t i o n t o the argument i n the
For based
i t , l e t us
literature
on
15
2.2
Definition As
based
pointed
on C o h e r e n t
out
in
the
Vorticity introduction,
the
widespread
a c c e p t a n c e of the importance of c o h e r e n t s t r u c t u r e s substantial consensus most
amount
o f how
of
coherent
the
fitting
appealing
structures)
and
autocorrelation
remains c a n be
also
(visual being
Coherence
i s t o have
concept
analytic
i s defined either or
Many d i f f e r e n t
f o u n d b u t n e a r l y a l l o f them (e.g. pressure,
of enough
f u n c t i o n o f a random v a r i a b l e
o f some e n s e m b l e a v e r a g e .
quantities
s h o u l d be d e f i n e d .
t h i n g t o do
observations
theoretical calculations. the
b u t has y e t t o r e s u l t
a coherent s t r u c t u r e
researchers,
definition
research,
invited
are
based
t h e one
"A
turbulent vorticity Here, the
structure
fluid over
average
of
average
over
structure Vorticity vorticity
that
the
momentum o r v o r t i c i t y )
associated
a s an
with
a
"phase c o r r e l a t e d " structures
different to
survives
Oc and
for
We
will
example:
phase
scale
correlated
i t s spatial extent."
similar
refers
the
dynamical
l a r g e s c a l e r e f e r s to s c a l e comparable
s h e a r f l o w and
the
definitions
i s a connected, large
mass
For
on
by H u s s a i n ( 1 9 8 3 )
coherent
in a
through as
w i t h the N a v i e r - S t o k e s e q u a t i o n or i t s d e r i v a t i v e s . choose
a
phases. its
stage
such
of
refers to the
The of
the
same p h a s e ,
phase
of
development
averaging
the other part
t o the e x t e n t of
is called
is
a
ensemble not
the
coherent
( i t s age).
called
coherent
i n c o h e r e n t (random)
16
vorticity Br. In p r e s e n t i n g
this definition,
i t i s n o t my i n t e n t i o n
a r g u e how c o h e r e n t s t r u c t u r e s s h o u l d the
scope of t h i s t h e s i s t o e n t e r
This
definition
be d e f i n e d .
this
to
I t i s beyond
far-reaching
argument.
i s chosen because of t h e r e l a t i o n between t h e
d e f i n i n g q u a n t i t i e s (0 a n d CJ) a n d
the
The
r e c o g n i t i o n b a s e d s o l e l y on
first
point
t o note i s that
the above d e f i n i t i o n knowledge
of
isdifficult
time
ensemble average t o i d e n t i f y method
must
instantaneous
or l o c a l
then
the
any
refine
requires
Our
a l l o w us t o d e f i n e
function
of
be
the
An o p e r a t i o n a l l y
field(s),
There should
i n the f i r s t
the
rotational cylindrical
and
n o t be
place.
considerations,
the
major
energy E r .
the
first
Er
is a
fluid
t h e r a d i u s R and a n g u l a r CJ i s
i t would
constant be
noted that
In
column,
v e l o c i t y CJ.
throughout
natural
s t r u c t u r e s t o f i t our p r e v i o u s should
of
t h e s t r u c t u r e f r o m an
structures afterwards.
the
of
column o f f l u i d , Therefore,
initial
and A n g u l a r V e l o c i t y
being
approximation
some
result
time averaged parameter
model i s b a s e d on e n e r g y
variable
the
the s t r u c t u r e s .
preferred s p a t i a l averaging
2.3 V o r t i c i t y
as i t r e q u i r e s
wordings.
t h e c o h e r e n t s t r u c t u r e s t o p e r f o r m t h e ensemble
a v e r a g e , a n d a t t h e same
easier
definition
to
define
model u s i n g
f o r t h e above f l o w
the
CJ.
F o r such a structure.
the
coherent
Moreover,
i t
structure, vorticity
17
and a n g u l a r
velocity
Analytically,
are
actually
proportional
(ft = 2u>) .
i f we c o n s i d e r an eddy i n i t s CM f r a m e , we c a n
d e f i n e u> a s V
(2.2)
= wxR ;
where R i s t h e r a d i a l v e c t o r parameters
are
defined
independent of t h e i n e r t i a l curl
from in
the
the
frame
CM.
CM
{Note
frame
and
that
the
thus
are
of t h e observer.} Taking the
o f t h e e q u a t i o n , we have
(2.3)
R = VxV = Vx(wxR) = 2u> - R ( V « C J ) + (R.V)w .
For our r i g i d l y the 2D
second case,
rotating
and t h i r d
fluid,
u> i s
constant
everywhere
t e r m v a n i s h a n d fl = 2u>.
and
u
are
perpendicular
to
V
and R, s o t h e s e c o n d t e r m v a n i s h e s .
the
z-axis
which
is Our In
c e n t r a l p a r t o f t h e v o r t e x m o t i o n , R i s s m a l l a n d we
also
good
be s i m p l i f i e d
the
i n t o ft = (2 - ~R>V)u.
expect the s p a t i a l a
along
In the general
ft
equation can f u r t h e r
both
so
r a t e o f c h a n g e o f w t o be s m a l l ,
r e p r e s e n t a t i o n f o r ft. M o r e o v e r ,
by p a r t i a l
spatial
instantaneous v o r t i c i t y must interpolated
derivatives
b u t ~£>
vorticity is
t h u s co i s i s defined
defined
by
the
v e c t o r s V a n d R; t h i s means t h a t c a l c u l a t i o n o f involve fields
finite
difference
(unless
vorticity
measured) which a r e secondary
interpolated
schemes
on
can
directly
be
values.
This
some
is
bound t o be l e s s a c c u r a t e t h a n t h e p r i m a r y i n t e r p o l a t e d v a l u e s of R a n d V w h i c h c a n be f o u n d d i r e c t l y
from t h e t r a c e r
paths.
18
In
other
words, t o f i r s t
order a p p r o x i m a t i o n around the
c e n t e r o f a v o r t e x m o t i o n , we c o u l d u s e t h e v a l u e s o f co a s indicator the
f o r 0.
T h i s b r i n g s t h e two s i d e s o f t h e e n e r g y a n d
momentum a p p r o a c h
structures
an
t o g e t h e r a n d we hope t h a t
s o r e c o g n i z e d c a n be u s e d w i t h b o t h
the
coherent
approaches.
2.4 The D e f i n i t i o n The
above
recognition definition
summarizes
i s based of coherent
"A c o n n e c t e d , minimum which
on
structures
contour
use of a n g u l a r speed
in
o u r 2D s i t u a t i o n
velocities.
coherent
Four c o h e r e n t
fluid of
mass o u t l i n e d
angular
speed,
i n the d e f i n i t i o n . identified
from a s p a t i a l
A
actual
by t h e within
plot
diagram
showing
definition
o f co i s shown i n F i g . 2.
the
field
of
angular
radius
R
and
constant
e x p e c t t h e v e l o c i t y a s V(r)=cor up t o 0^r)
(7.4)
Equivalently, an e q u a t i o n
Taking
.
upon f i n d i n g
S, C a n d a, we c a n e x p r e s s
The
find
cos = 0.64 ± 0.08
To
t h i s as
f o r cos0:
S = 0.2, C = 0.59 a n d a = 4,
= 34°.
The
o f C a n d , we g e t
c o s 0 = Tr / (8SCa) .
(7.5)
we
S i n term
to
must
believe
be
a
the
proposed
transition
model.
between t h e two
mechanisms even i f t h e p r o p o s e d model a c c u r a t e l y d e s c r i b e s t h e present
grid
infinity, Von
Karman
system.
we
return
mechanism
understanding,
the
This to
is
the
should model
single prevail.
should
v a l u e s of a t o see the r e l a t i o n mechanism and t h e geometry.
because
of
be the
when
bar
a
approaches
situation
To tested
gain with
structure
a
and t h e deeper
different production
1 03
S t r u c t u r e S i z e and E n e r g e t i c s
All
structure
dimensional which
is
tangential equal
forms. a
rigidly
With
its
circular
the
respectively
the
with
rotating
t o the towing
exception
energy, every
and
normalized
of
structure
in
the
the
towing
non-
structure with
outer
diameter
CM
velocities
parameter
D
and t h e
is
compared
structure.
energy
The
are
speed and t o t a l
CM
compared
energy of the
structure. Amongst a l l t h e s t r u c t u r e p a r a m e t e r s d i r e c t l y
the major u n c e r t a i n t y i s t h e parameter
is
highly
around a s t r u c t u r e . the
to
i s associated to this
ideal
translational
eddy
speed and
No t r a n s l a t i o n
counterpart
velocities
are
i s done by u s i n g t h e i d e a l
t o t h e mesh w i d t h .
translational
ideal
This
s p e e d Ug e q u a l
structure.
with
parameters
boundary
can
and
dependent
be
defined
is
low,
may
been drawn by h a n d .
situations.
structure.
This
on t h e a c t u a l t r a c e r d e n s i t y density
clearly.
the
is
high,
However, f o r p l a c e s
boundary
defined
is
less
f r o m what w o u l d have
T h i s i s not j u s t a problem of t h e package
process
the boundary a l s o
the
be q u i t e d i f f e r e n t
as t h e f l o w f i e l d a t such recognition
of
F o r p l a c e s where t h i s
where t h i s d e n s i t y predictable
size
calculated,
places
is
doubtful.
The
of assuming the longest v i s i b l e
cannot
However,
be
completely
justified
i n using photographic
manual
s t r e a k as in
such
techniques,
much
1 04
higher not
t r a c e r d e n s i t i e s c o u l d be u s e d and
apparent.
In
t h e new
t r a c e r c o n c e n t r a t i o n and e v e n l y over places
i t i s hard
the viewing area.
where
uncertainties
the in
go
back
parameter recognition of
the
the
reasonably
structure
in
not
These
This
expect
values
i f we
the
In
are
similar
consistently "actual"
our
and
have
we
have
summed
local
conditions
and
i f we
the
the
we
can
still
size
mostly
over
used d i f f e r e n t
and the
areas.
situation
is
i n t h e r a t e of c h a n g e o f This to
is
because
remain
roughly
structures
s h o u l d not
Assuming
the
velocities
average
densities and
this
take
were
However, the
vary.
l o o k a t an a r e a o f a f i x e d
value,
we
the exact v a l u e .
tracer
or not
recognized structure
average
a r e more i n t e r e s t e d
structure.
average
high
tracers
to c a l c u l a t e the
recognized
E q u i v a l e n t l y , we
r a t i o to that
rates
v a r i o u s p a r a m e t e r s a r e n e a r l y u n i f o r m and an
low
cannot
study,
of
they would vary
c o n s t a n t d u r i a g our experiment under
is
a r e bound t o h a v e is
We
f o r every
evolution
the parameters r a t h e r than we
the
whether
inability
seem v e r y d i s s a t i s f a c t o r y .
so bad
a very
parameter,
check
estimated.
accurately
the
s t r u c t u r e a r e a and T h i s may
to d i s t r i b u t e
density
and
a l s o poses o t h e r problems.
energies.
c a n n o t use
is a critical
plots
for granted.
interested
problem
interpolation.
to
is
the
T h e r e f o r e , we
information
F o r s t u d i e s where s i z e to
s y s t e m , we
so
of
can
estimate
of
the
c h a n g e of
the
be d e s c r i b e d
by
the
rates
of
105
evolution. By l o o k i n g a t t h e p l o t s structures
at the i n i t i a l
found t h e average
initial
and
selecting
well
t i m e s ( t ^ 12/30 s o r x < 1.6M), we diameter ( c a l c u l a t e d
area
by
of the s t r u c t u r e s
was
circular
a p p r o x i m a t i o n ) o f t h e s t r u c t u r e , D, a s
(7.6)
D = 0.8 ± 0.1 mesh w i d t h s . The
also of
found.
boundary
This
s p e e d , Um , 0
speed
boundary
over
speed.
structure
(7.7) where
from
i s c a l c u l a t e d by c o m p a r i n g t h e mean v e l o c i t y
the recognized structure with
mean
the
initial
recognized
the
that
o f t h e i d e a l eddy.
ideal structure
The
i s 2U/3 where U i s i t s
T h e r e f o r e , we e s t i m a t e t h e b o u n d a r y
speed
of
by t a k i n g
Urn = 3/2 * U' ; U' d e n o t e s t h e a v e r a g e s p e e d o f t h e s t r u c t u r e .
considerations estimate
of
on
root
mean
squared
speed
also
Similar gives
Urn. M o r e o v e r , we c a n c a l c u l a t e t h e s p e e d
an
either
from t h e g r i d of t a n g e n t i a l speed o r from t h e g r i d of v e l o c i t y components. the in
tangential agreement
(7.8) in the
T h e r e f o r e , we have f o u r
Um
0
speed of t h e s t r u c t u r e .
with
estimates
of
They a l l t u r n e d o u t be
e a c h o t h e r a n d we f o u n d
= (0.28 ± 0.08) Ug
a l l samples. tangential
particularly
different
The a g r e e m e n t speed
good
as showing t h e
and
between
velocity
(usually within
velocity
v a l u e s g e n e r a t e d from component
5 % ) . T h i s was
interpolation
is
stable
grid
is
interpreted with
the
106
present (about
routine
and
data
density.
The l a r g e r
15%) b e t w e e n r e s u l t s f r o m t h e a v e r a g e
discrepancy
speeds
and
r o o t mean s q u a r e d
s p e e d s was i n t e r p r e t e d a s t h e d e p a r t u r e
the
rotating
ideal
rigidly
Although
of
be
This
than
from t h e v e l o c i t y
grids
should
be u s e d a s a d i r e c t
t r a n s l a t i o n a l e n e r g y o f t h e s t r u c t u r e s was of magnitude l e s s than
The c a l c u l a t e d the
interpreted the
be
estimate
of
i n two ways.
the
stationary fitting
m o d e l t h a t Vcm=0.
interpolated
an
internal
This
i t i s the basic
grid
t o g i v e us back t h e a s s u m p t i o n .
just
i s much
calculation.
Firstly,
consistency
found
the r o t a t i o n a l
t r a n s l a t i o n a l energy
uncertainty
s u r p r i s e f o r an
model as
directly
value c o u l d then
3-4 o r d e r s
energy.
no
situation.
Um. The
in
from
i t was n o t done i n t h e p a c k a g e , c a l c u l a t i n g t h e
boundary v e l o c i t y trivial.
the
to
or t o t a l smaller can
be
assumption
T h e r e s h o u l d be
generated
under
this
T h i s c a n be c o n s i d e r e d
check
and
t r a n s l a t i o n may be smeared o u t i n t h e f i t t i n g .
the
actual
However, based
on o b s e r v a t i o n s we b e l i e v e d t h a t t h e t r a n s l a t i o n s a r e a c t u a l l y small.
Considering
(Hussain
1 9 8 3 ) , i t was
packed
at production
state physics). translation.
the
s t r u c t u r e s t o be m u t u a l l y e x c l u s i v e
found
quite
closely
(not i n t h e sense of c l o s e - p a c k
in solid
There j u s t
From
the
that
they
i s n o t much
plots
of
are
space
available
for
r e c o g n i z e d s t r u c t u r e s , we
l o o k e d a t t h e CM p o s i t i o n s a t s u c c e s s i v e t i m e s .
We f o u n d
that
107
most s t r u c t u r e CM's a r e p r a c t i c a l l y
stationary
with
supports our a n a l y s i s
the
streak
motions.
This
when
r o t a t i o n a l m o t i o n s a s shown by t h e s t r e a k s
are
than
by
the
structure
translations
shown
compared
much the
that
greater structure
centers. We e x p e c t t h e t o t a l e n e r g y t o be limits total Fig.
of u n c e r t a i n t i e s . energy
17.
The
f o r the total
rotational
A p l o t of r o t a t i o n a l energy recognized
energy
structures
i s calculated
e n e r g y i s c a l c u l a t e d by a r i g i d shown
i n section
moment o f i n e r t i a angular v e l o c i t y (7.9)
a)' = I
(4.4),
body c i r c u l a r
we f i r s t
I by d e f i n i t i o n . defined
3
3
against
i s shown
in
The r o t a t i o n a l approximation.
c a l c u l a t e d the structure With
a
revised
average
as
{V /( |R- -Rcm| ) } / N . L
L
The r o t a t i o n a l e n e r g y o f t h e s t r u c t u r e ( 7 . 10)
the
f r o m d e f i n i t i o n by
summing t h e e n e r g y c o n t e n t a t e a c h g r i d p o i n t .
As
within
i s c a l c u l a t e d as :
E r = (ICJ' ) / 2 . 2
This i s n o t t h e average a n g u l a r v e l o c i t y c a l c u l a t e d from t h e a n g u l a r v e l o c i t y g r i d , we h a v e done s o many s m o o t h operations on t h e u> g r i d t h a t i t i s n o t a d v i s a b l e t o u s e i t f o r a n y t h i n g other than s t r u c t u r e r e c o g n i t i o n .
108
Figure
17 - P l o t of c a l c u l a t e d r o t a t i o n a l e n e r g y v s energy f o r r e c o g n i z e d s t r u c t u r e s .
total
109
key : — — — ideal profile without mixing possible profile with mixing E{ by definition
UJ' is defined so that
the two shaded
parts
Figure
18 - S p a t i a l
have same
p l o t s of w ( r ) and V(r) situations.
area
for different
110
key :
ideal profile without mixing possible profile with mixing E{
by definition
rigid body profile E
Figure
r
by
19 - S p a t i a l p l o t o f V ( r ) situations. 2
I(jJ' /2 2
for different
111
Experimentally, determined energy.
again
To
consistently greater
1D
the
realistic
profile
with
r a d i u s of t h e
calculate
sketched
An
symmetry, E circular R'CL)'.
the
2
2
w o u l d be with
To
calculated
energies
areas
the
the
proportional
t o the
When t h e m i x i n g be
greater
two
parameters
Note
R'
and
t h a n E,. values.
are
a
more
notice that (R
to
0
The
as
fluid.
We
by d e f i n i n g
p l o t as
in
f u r t h e r assuming rigidly
can the the
radial
rotating
boundary t a n g e n t i a l speed
i d e a l eddy i s a l s o shown i n
discrepancy
and
the
E ,
a plot
2
between the
the d i f f e r e n c e i n the that
the
R')
approximately
r e l a t i o n b e t w e e n E,
f o u r t h power o f
l a y e r has
the w(r)
e n e r g y of a
s u c h an
w o u l d be
plot.
we
possible
shows
s t r u c t u r e w i t h the
areas
total
radius
clarify
i d e a l and
line
First,
r a d i u s and
i s shown i n F i g . 1 9 .
in
solid
Consider
2
d e f i n e d under t h i s d e f i n i t i o n
shaded
the
between
E .
cj(r) from the V ( r ) p r o f i l e
V ( r ) p r o f i l e of
figure.
of V ( r )
ideal
this modified
eddy
The
e n e r g y by
by d r a w i n g a l i n e a c r o s s
With
relation
structure i s increased
a v e r a g e w'
f i g u r e so t h a t t h e two same.
The
so
denote the c a l c u l a t e d
oo p l o t o f t h e
mixing.
of the
t h e new
OJ( r ) =V( r ) / r . be
linear
rotational
s p a t i a l V and
introduce mixing
can
the
energy
than the c a l c u l a t e d t o t a l
i s a strong
shown i n F i g . 18.
effective
rotational
i n t e r p r e t t h i s d i f f e r e n c e , we
e n e r g y by E, and
situation
we
found that the
However, t h e r e
them. total
is
we
two
rotational
r a d i u s f o r an
shaded
energy i s
ideal
grown t o a c e r t a i n l e n g t h ,
two
E
eddy. 2
T h i s e x p l a i n s the d i f f e r e n c e between
will the
1 12
From t h e variations This
is
above
originate caused
by
consideration,
we
note
that
f r o m t h e c h a n g e of v e l o c i t y p r o f i l e V ( r ) . outward
energy d i f f u s i o n or mixing.
shown i n t h e f i g u r e , t h e two e n e r g i e s a r e v e r y related. (or
a
This family
structures. physical the
is of
This
evidence
that
profiles) should
mixing process
common
also
be
to true
t o be t h e same.
structures.
t h e o r e t i c a l models This
the system and a l s o
much
As
linearly
there e x i s t s a V(r) p r o f i l e
On
most as
of
mixing
expect other
the
hand,
e n e r g i e s c a n be
in
s h o u l d be done a s t h e n e x t
recognized
we
the
a b o v e r e l a t i o n between t h e two c a l c u l a t e d
used t o t e s t
a l l the
the
coherent
step in t e s t i n g
t o v e r i f y d i f f e r e n t m i x i n g models.
113
C?2 F i g u r e 20
O'A
f^-
016
- L o g - l o g p l o t of i n i t i a l d e c a y r a t e A as f u n c t i o n of the s t r u c t u r e r a d i u s R
a
1 14
0,2 F i g u r e 21 - P l o t o f
0,4 initial
0,6 decay r a t e A vs
0,8 1/R
2
11 5
7.1.2
S p o n t a n e o u s E n e r g y Decay R a t e of C o h e r e n t According
the
by
(7.11) where
and
A = 16u v
(7.12)
is
/ R
the A
2
structure i s
;
kinematic
v i s c o s i t y , R i s the
i s d e f i n e d through the
r a d i u s of
the
equation:
E(t) = E(t=0)exp(-At) .
i s b a s e d on a s i m p l e
u s i n g c o n s i d e r a t i o n s of objective
of
the
d e r i v a t i o n from a c i r c u l a r power
contrasted
to i n t e r a c t i o n
initial
t r a n s l a t i o n , we
dissipation.
experiments
"Spontaneous" here r e f e r s to
the
Loewen,
:
s t r u c t u r e and
In
Ahlborn
s p o n t a n e o u s e n e r g y d e c a y r a t e A of a c o h e r e n t
given
This
t o t h e m o d e l p r o p o s e d by
Structures
study
of
f l o w and the
is
this
structure-fluid
with
period
to
It
structure another
relation.
interaction
as
with other s t r u c t u r e s .
system
when we
e x p e c t t h i s mechanism t o dominate
have
little
the
energy
d e c a y of t h e c o h e r e n t s t r u c t u r e s . As
discussed
difficulties and
every
recognized the
are
the
i n determining structure.
region
on
has
i n t e r a c t i o n s by
section,
size
we
the
streak data
sufficient We
can
judging
still
(hence r a d i u s )
have t o c h e c k t h e
size determination.
other
last
the
We
plot overlayed
boundary
accurate
in
for
each
result with
t o judge
information density
for
there
l o c a l c o n d i t i o n s of
the
structure.
With these
considerations,
we
select
well
recognized
structures
free
i n t e r f e r e n c e by
other
of v i s i b l e
can
the
whether
a l s o check whether
the
have
1 16
structures
or
maintaining the
the
flow,
i . e .structures
that
more o r l e s s t h e same g e o m e t r y .
structures'
decay r a t e a g a i n s t
approximation) i s given structures
i n F i g . 20.
i n t h e same r u n f a l l
decay
A l o g - l o g p l o t of
t h e i r radius We
on a
while
(by c i r c u l a r
found that p o i n t s
single
straight
M o r e o v e r , t h e s e l i n e s a r e f o u n d t o o f t h e same s l o p e
from line.
around
-2
although the intercepts are d i f f e r e n t . This
shows
that
the
energy
proportional
t o the square of t h e
within
same
the
run
as
p r o p o r t i o n a l i t y constant calculated
v
value
of
the
system.
established
later.
known surface
by
(7.11).
over
numerical
contaminants experiments
bulk were but
From t h e
found
not the
cm /s.
model,
viscosity was
about
than the
We a l s o
found
f o r experiments that
were
2
i n t e r p r e t e d as t h e combined e f f e c t of
surface
v i s c o s i t y several
To
a n o t h e r p l o t of the
value
v i s c o s i t y o f 0.01
is
free
corresponding
runs.
an e s t i m a t e o f t h e k i n e m a t i c
u's were h i g h e r
This
a
structures
However, t h e
different
i s shown i n F i g . 2 1 .
2
c o n t a m i n a n t s and i n h e r e n t that
the
i s an o r d e r o f m a g n i t u d e h i g h e r
water bulk
the c a l c u l a t e d
surface
of
predict changes
The
which
2
done
1/R
should give
0.2±0.1 c m / s
that
radius
the p r o p o r t i o n a l i t y constants,
decay r a t e a g a i n s t this
decay r a t e A i s i n v e r s e l y
surface
with
viscosity.
contamination
order of magnitude g r e a t e r viscosity considered
(Criddle at
proportionality
the
1960) time
It
is
c a n have a than
the
Surface of
of the decay r a t e
the with
1 17
respect
t o 1/R
2
experiments with skimming
i s quite better
the surface
evident
study
contaminants (e.g.
the
relation
we
weakness.
Although
i s considered
physical
the present routine
p r o d u c e i n t e r p o l a t e d f i e l d s much d i f f e r e n t
from
physically.
information
in this
such
equations
This
as
c a n be i m p r o v e d by p u t t i n g
interpolation.
limit
in
selecting
a
sufficient
time
governing
lower accuracy
of
analysis
sampling and
the rates
curvature
T h e s e c a n be o b t a i n e d
time
decreasing
and
experiment. software
is
may
rate.
velocity.
by
realistic be
The to
to
control
and
fields.
e a s e d by s o f t w a r e package
define
increasing
the
other
of change of t h e
the sampling rate at l a t e r
The a b i l i t y
I n c l u s i o n of
incompressibility
v a r i a b l e s c a n be u s e d t o i n t e r p o l a t e more The
CGRID1 h a s some
constraints
differential
as t h e major
are s t i l l
expect
more p h y s i c a l
the
System
p r o v e n t o be a d e q u a t e i n most s i t u a t i o n s , t h e r e
what
of
and Recommendations
grid interpolation routine
pictures that
Further
of v i s c o s i t y .
7.2.1 L i m i t a t i o n s o f t h e P r e s e n t
software
plots.
b e f o r e e a c h e x p e r i m e n t ) s h o u l d be r u n t o
decay r a t e as a f u n c t i o n
The
the
c o n t r o l l e d surface
t e s t t h e h y p o t h e s i s and a l s o t o
7.2 D i s c u s s i o n s
from
the the
requires angular exposure
times i n the
exposure
time
by
a l s o one o f t h e m a j o r a d v a n t a g e s o v e r t h e method
1 18
of p h o t o g r a p h i c
flow
visualization.
Theoretically,
s t u d y s t r u c t u r e s w i t h s c a l e t i m e s much l a r g e r flow
drifting
model
c a l c u l a t i n g the g r i d interpreted
as
primary data. where Vcm higher
system
being The
in
the
package
with
caused
than t h a t of
the
little
has
been t r i e d
success.
major We
applicability
of t h i s model
in
situations
f u r t h e r evaluated with data
limitations
in
our
present
the range
of towing v e l o c i t y
t h e r e s u l t s , a r e a l l d e p e n d e n t on t h e the primary created
system
h a v e shown t h a t t h e s i z e of v i e w ,
analysis,
images.
much
The
hazard
noise level
in extracting
and
the
of
present
system
t h i s was
first
in
package
the
priority
over
of
resolution
of
spatial
of o u r d i g i t i z e d
160
i s a s u b s t a n t i a l c o s t f o r the u s e r s .
As
t o be g e n e r a l ,
real
UBC,
complete 250
running
possible
(3-4
of
interest is at
normal
s e c o n d s of
flow)
corresponding
Running the system
t o be
optimization
package
dollars,
w o u l d n o t be
t h e p a c k a g e has
the
set of data
computer
dollars.
the p r e s e n t environment of
in
At
r e q u i r e s more t h a n
pictures
t h i s primary information.
c a n be done i f t h e e x p e r i m e n t
a
duration
required
designed
sharply defined.
are i n
the accuracy
M o r e o v e r , t h e l a r g e amount of c o m p u t e r r e s o u r c e s
version
is
precision.
hardware.
around
This
in
by t h e l a r g e u n c e r t a i n t y i n t h e
i s n o t z e r o s h o u l d be
The
the
can
pictures. The
of
we
to
extensively in
recommended.
A
working
i m p l e m e n t e d i n a much
lower
119
cost
environment. These
a r e a s s h o u l d be
improvement
over
the
given
present
highest system.
p r o p o s e d ways t o a d d r e s s t h e d a t a straightforward
way
is
v i d e o r e c o r d e r and t h e equipment.
This
priority
problem.
t o r e p l a c e the equipment,
involves
a
with
larger
more
Paven
possible
time.
Two
up-to-date
amount
evaluations
of
capital in
the
(Dewan e t a l ,
e t a l 1985) on how t h e s y s t e m c o u l d be i m p r o v e d w i t h i n a
d e f i n e d budget
have
microcomputer
and
been VCR
o t h e r recommendations microcomputer Judging
photographic
carried system
are
involving
set at consist
quite attractive.
setting
the
limited
streak
up
i n video d i g i t i z a t i o n ,
technique
has
been
a
Their
stand
frequency.
in this
alone
a
section.
return
to
pictures
of
the
system
l i g h t and a s t r o n g s t r o b i n g The
the
p r o p o s e d by S. Loewen.
resultant
pictures
In are
light would
s t r e a k s s u p e r i m p o s e d by i n t e n s e s p o t s s h o w i n g t h e
t r a c e r p o s i t i o n s as a f u n c t i o n of t i m e . can
recommended
view s i z e of the p r e s e n t system
known of
the
on
w i t h a dim background a
and
be d i s c u s s e d
t h i s technique, time-exposured taken
out
environment w i l l
and t h e n o i s e l e v e l
we
The
namely t h e
i n v e s t m e n t b u t w o u l d e n a b l e t h e s y s t e m t o be w o r k a b l e shortest
any
T h e r e have been two
acquisition
microcomputer,
for
have tracking
a
higher i s nearly
tracer
density
W i t h such
a
system,
s i n c e t h e p r o b l e m of
s o l v e d by t h e b a c k g r o u n d
lightning.
1 20
T h e r e a r e many a d v a n t a g e s area
is virtually
the photographic t h a n any for
i n such a system.
u n l i m i t e d because the s p a t i a l
image i s s e v e r a l o r d e r s o f
e x i s t i n g video d i g i t i z e r .
the
same
area,
s h a r p l y than at different
thereby
present.
portions
of
of
Moreover,
area.
We
parameters
can
also then
more
digitize superimpose
visually
choose
s t r u c t u r e s f o r study r a t h e r
S u c h d i g i t i z a t i o n s a r e s u p p o s e d t o be done by
hardware
c e n t e r and
the r e s u l t s
transferred
considering
the
difficulties,
such a system
can be u s e d
MTS,
the
tracers
have
to
the e x p e r i m e n t a l
t o s t u d y may
more s e v e r e l y t h a n t h e p r e s e n t s y s t e m .
t h e i r own
to
minidisks.
during which
The
primary
do n o t
digitization.
limited
reason
ones.
h e l p a b i t but t h i s
is than
I t w o u l d be
overlapped tracers with larger
s t r o b e s o f d i f f e r e n t c o l o r s may be
be
time
move o v e r d i s t a n c e s g r e a t e r
s i z e s t o be t e m p o r a l l y r e s o l v a b l e .
to d i f f e r e n t i a t e
still
higher
viewing
In
to
the
can a l s o
of
small
tapes or
has
resolution
smaller tracers
on t h e l u c k o f t h e outcome i n a
in the computing
that
use
we
viewing
magnitude
a s i n g l e p i c t u r e and
g o o d t r a c e r d e n s i t y and
than depending
can
defining
them back i n t o c o m p u t e r memory. areas
We
The
hard Using
assumption
r i g o r o u s l y t e s t e d by p r e l i m i n a r y e x p e r i m e n t s . know
how
The
i n c o r p o r a t e t h i s new a major d i s a d v a n t a g e .
superimposed
software
that
colors has
to
turn be
method i n t o t h e p r e s e n t one The
author
expects
out
We upon
developed
to
also
presents
such a l i n k
between
121
the
two A
the
systems would take more
be
c a r r i e d out
belief
that the present
where
we
can
experiments
do
continually
trying
Further and
should
of
system It is
using
the
data
system
the
technique.
the
Apple
flow
acquisition rather
computer spatial
than
is
not
resolution
I recommend a m i n i m a l u p g r a d e of
experiments.
recorder,
Always t r y i n g
e v e n t u a l l y render
this
my
point
turbulent
the
i f p o s s i b l e , the v i d e o
more
s t a t e - o f - a r t may
and
the then
t o push t o
system o b s o l e t e
the
before
utilizing i t . Both proposals
has
upgrading
with
such
been d e v e l o p e d t o t h e isotropic
t o i m p r o v e on
needed.
in
ourselves.
a microcomputer with higher
m i c r o c o m p u t e r and
fully
2D
start
development
definitely
doing
committing
s y s t e m has
minor
We
start
operational.
subsequent gains
before
elegant
with
equipment.
realistic
fully
d e t a i l e d e v a l u a t i o n w i t h p r e l i m i n a r y a n a l y s i s of
a n t i c i p a t e d p r o b l e m s and
should
is
months t o be
r e q u i r e the e x t e n s i v e a n a l y s i s system
been d e v e l o p e d i n t h e p r e s e n t
system.
We
have
to
that deal
w i t h the a n a l y s i s c o s t problem s e p a r a t e l y .
I t was
recommended
that
in
stand
the
system
microcomputer However, a t t h e was
considered
proposal.
should
be
implemented
environment(Dewan et a l ,
Pavan et a l
t i m e of e v a l u a t i o n , t h e MTS as
S i n c e UBC
the
a
t h e FPS
Array
1985).
mainframe computer
o n l y a l t e r n a t i v e to the
installed
alone
microcomputer
Processor
(AP),
1 22
this
s h o u l d be a d i f f e r e n t
cost of running change
the
story.
The m a j o r c o n c e r n
t h e s y s t e m c a n be s i g n i f i c a n t l y
dollars
as
lowered
s y s t e m t o r u n on t h e a r r a y p r o c e s s o r .
e n v i r o n m e n t , one h o u r o f e x e c u t i o n contrasted
time
costs
t o t h e 24 d o l l a r s
over t h e i f we
I n t h e MTS
480
computer
f o r t h e AP.
In our
s y s t e m , t h e memory u s a g e c o n s t i t u t e d more
than
the
i s f r e e on t h e AP.
overall
Converting solution
computing
and t h i s
t h e system t o use t h e
to the cost
As
charges
the
AP
seems
one-third
to
be
an
of
ideal
problem.
present
package
i s written mostly
FORTRAN w i t h a few MTS FTN e x t e n s i o n s
for file
i n standard
manipulation,
translating
i t t o t h e APFTN64 l a n g u a g e u s e d i n t h e AP s h o u l d
be
Except
trivial.
routines
called
f o r the are
also
c o n v e r s i o n c a n a l s o be d o n e . a way t h a t t h e n u m e r i c a l graphical without AP to
analysis.
visible
graphics written
FORTRAN,
t h e two p a r t s
changes seen from t h e o u t s i d e .
With
We c a n u s e t h e
of
the
i n t h e image p r o c e s s i n g . be
able
personally
dislikes
we
mainframe
out still while
A summer o f an to
w h o l e c o n v e r s i o n and make t h e s y s t e m f i n a n c i a l l y author
such
completely
such a c o m b i n a t i o n ,
APSC 459 t y p e p r o j e c t i n UBC s h o u l d
The
similar
p r o c e s s i n g and pass t h e r e s u l t
have t h e e x t e n s i v e g r a p h i c s s u p p o r t cost
system
The p a c k a g e was w r i t t e n i n
We c a n s e p a r a t e
MTS f o r g r a p h i c s p l o t s .
most
in
most
data p r o c e s s i n g i s independent of t h e
t o do a l l t h e n u m e r i c a l
reducing
package,
finish
the
feasible.
the idea of downloading
123
the
system
further
to
a
microcomputer.
development
of
the
In
such
system
an
i s hard.
microcomputer
systems a r e upgraded c o n s t a n t l y ,
considerable
i n v e s t m e n t f o r each improvement
and hardware.
In u s i n g a system t h a t
the
center,
computer
immediate frequent
attention
we if
can
i t would
exchange.
The
also
make
us
stand
s p e e d and l i b r a r y
wrong
alone
from
their
and a l s o
in
from
A stand
information
s u p p o r t i n a mainframe
a l s o c a n n o t be m a t c h e d by any m i c r o c o m p u t e r
a
s u p p o r t e d by
benefit
goes
be
in both software
i n t e r a c t i o n with these knowledgeable people.
a l o n e s y s t e m may
out
A l t h o u g h most
is directly
always
anything
environment,
system
and t h e s e may
turn
t o be a g r e a t h i n d r a n c e f o r d e v e l o p m e n t .
7.2.2
E x t e n s i o n t o a 3D Although
System
considerations
for
possible extension
s y s t e m were a l w a y s made d u r i n g t h e d e v e l o p m e n t such
extensions
a r e by no means t r i v i a l .
t h i s s y s t e m c a n be c a r r i e d o v e r , b u t t h e data the
c a p a c i t y of
equipment
most
upgrading,
of
our
existing
successful
3D
of t h e
system,
The m a j o r
ideas in
increase
d e n s i t y , a c c u r a c y and memory s i z e w i l l
to a
demand
definitely
equipment.
extension
will
in
exceed Without
be
very
improbable. Fundamentally, the d e f i n i t i o n in
t h e 2D
very
s y s t e m must be m o d i f i e d .
of coherent s t r u c t u r e s T h i s r e l a t e s back
a r c h i t e c t u r e of the present system.
to
used the
Generally speaking,
1 24
we s t a r t e d by c h o o s i n g our
i d e a l coherent
parameter
from
estimations,
a property
structure.
physical we
finding
the
step t e l l s defining
peak
We t h e n
that
the
throughout
v a r i a t i o n of the Through
real
such
operation
of
s t r u c t u r e i n v o l v e s t h e two s t e p s o f
and
2) o u t l i n i n g
us where we s h o u l d
the boundary.
The
l o o k a n d t h e s e c o n d one
1)
first
i s the
operation.
In
2D,
candidate. Because
angular I n 3D,
of
general
velocity
we
have
is
to
case.
to
We
simplifications.
find
will
a
a
probably
to
be
and
of
so
general
there
to
o f 3D f l o w s ,
look
consider
f o r the for
only
some flows
z-component v e l o c i t y i s
interest
should
variable.
candidate
t h e component on t h e x y - p l a n e .
are
d i s c i p l i n e s and
i.e.
a suitable
similar
have
F o r e x a m p l e , we c o u l d
much s m a l l e r t h a n
The
find
suitable
t h a t a r e n e a r l y two d i m e n s i o n a l ,
usefulness
found
t h e i n c r e a s e d v a r i e t y and c o m p l e x i t y
we may n o t be a b l e
exist
infer
considerations.
proposed
i d e n t i f y i n g a coherent
that i s constant
be
Such
f o r many no
doubt
systems
different as
to the
of such s t u d i e s . defining
parameter
need
not
be a n g u l a r
velocity,
h o w e v e r i t must be a p a r a m e t e r t h a t c a n be c a l c u l a t e d d i r e c t l y from t h e t r a j e c t o r y . vorticity the
full
angular
is
not
angular velocity
F o r t h e same r e a s o n recommended.
velocity
t h a t we have i n
One p r o b a b l e
vector.
The
2D,
c h o i c e w o u l d be
magnitude
v e c t o r seems t o be a r e a s o n a b l e
of
candidate
the as
1 25
the d e f i n i n g parameter. When we but
run
stricter
t h e e x p e r i m e n t , we
constraints
A s s u m i n g t h a t we better
digitizer,
developed
by
coordinate S.
have a we
as
more can
powerful use
the to
Another
in
color. in
the
resolution
of
determine the else.
fluid the
t r a c e r s and one
We overlap
have of
trace
image
and
v o l u m e of
I t w o u l d be
than the
to
capture
3
we
view,
its
and
consider
latter
will
levels
tracers.
Finding a neutral density
still
further limit
to obtain a high
successfully getting their
m a j o r p r o b l e m t h a t must be Once
we
can
fluid by
density
motion.
size nearly
image d i s t o r t i o n
The
fluid
tracer
t h e m e t h o d s of t h e
tracers at d i f f e r e n t
i n the
a neutral
The
will
also
everything
c o u l d e x p e c t a number d e n s i t y much
camera.
them
the
by
identified
internal
hence
proposed
across
be
get
when the
last and
of
smaller
chapter.
also
possible
viewed
by
one
number d e n s i t y
t r a c e r , p u t t i n g enough enough i n f o r m a t i o n coordinates
a
technique
even h a r d e r t o c o n t r o l the d i s t r i b u t i o n
c a l c u l a t e d by to
then
and
dimensional
method
have t o put
the
chapter.
microcomputer
possible
will
similar
stereoscopic
d e p t h s can
I n e i t h e r c a s e , we
tracer
the
The
to
last
Loewen i s t o p r o j e c t d i f f e r e n t c o l o r l i g h t
at d i f f e r e n t l e v e l s .
and
subjected
discussed
Sheu e t a l ( l 9 8 2 ) data.
are
i s the
of of
density first
solved.
such
data,
we
will
have t o f a c e
the
1 26
storage problem. form,
we
I f we c a n n o t t a k e d a t a d i r e c t l y
should
transform
possible.
A
three
resolution
256x256x256
them
to
dimensional and
a
this
image
form
pixel
further
64
This
should
be
c o o r d i n a t e s and a t t r i b u t e s possible. use
a
With
such
subroutine
manipulation
transformed
a
t h a t i s b e s t done i n a p i x e l
T h i s k i n d of data
reduction
s t o r a g e and c o m p u t a t i o n Hopefully
would
be
array
c e n t e r s and a l s o
should
joining
the present
of t h e s t r e a k c o n n e c t i o n We
then
i t was
drifting For
a
come
be
the
easy.
early
as
i t and
for
for
of
any
saving
can
With
phase w i t h o u t to
where
that
model,
we
have
present
the tracer
also
be
good p r i m a r y
s y s t e m t o be e x t e n d e d up t o
back
found
the
Locating
streaks
system.
direct data, the
many
difficulties.
the
original In
we end
coherent the
2D
the parameter e x t r a c t i o n w i t h the
model a r e l e s s a c c u r a t e 3D
of p i x e l
environment.
important
s t r u c t u r e d e f i n i n g p a r a m e t e r h a s t o be e x t r a c t e d . case,
data
time.
of the present
c o u l d expect
at
we w o u l d n o t h a v e t o do much s m o o t h i n g f o r t h e
algorithm
extensions
as
array
3D d a t a w i t h b e t t e r e q u i p m e n t , b u t e x t e n s i o n smoothing
gigantic
to a l i s t
virtual
with
levels
s t r u c t u r e , we c o u l d s o r t
simulate
list
array
(X,Y,Z,T,grey l e v e l )
a list
to
a
a
a s soon a s
grey
p r e s u m a b l y 30 f r a m e s p e r s e c o n d c o n s t i t u t e s structure.
in
than
the
stationary
model.
t o look at the flow a t d i f f e r e n t
d e p t h s a t t h e same t i m e a n d t h e s t a t i o n a r y m o d e l
may
not
be
1 27
able
to
d e s c r i b e the whole f l u i d
similar
fitting
methods,
the
field
adequately.
drifting
model
In u s i n g
seems
more
appropriate.
H o w e v e r , t h i s demands q u i t e an a c c u r a t e d a t a
set
as
to
the
we
have
trajectory. models.)
use
the
t h i r d order
(See A p p e n d i x A f o r t h e
If
this
proves
t o be
d i f f e r e n t methods t o e x t r a c t either the
not
fitted
fitting
major d i f f i c u l t y If
we
can
the
This
i n the
difference
t o o h a r d , we various
by p o l y n o m i a l s
trajectory).
t i m e d e r i v a t i v e s of
is
in
may
the
two
have t o
find
parameters
or u s i n g a n o t h e r considered
as
appears
to
problem
me
that
of
we
space
be a b l e t o e x t r a c t
kind
of
probably We routine
be o f l i t t l e
should
governing
Even
if
equations
incompressibility
consideration,
It
a high data d e n s i t y .
s i t u a t i o n s w o u l d be
r o u t i n e at
come t o
UBC
one
difficult.
that
does
does e x i s t ,
this
i t would
use. interpolation.
The
i n c o r p o r a t e v a r i o u s c o n s t r a i n t s g i v e n under
physical
into
we
interpolation.
need p h y s i c a l i n p u t t o d i r e c t t h e
different
coded
library
interpolation.
(3D)
cannot expect
Space i n t e r p o l a t i o n under such T h e r e i s no e x i s t i n g
second
extension.
a l s o s o l v e t h i s p r o b l e m and
t h i r d major
model f o r
the
the p h y s i c a l parameters w i t h c o n s i d e r a b l e accuracy, the
(i.e.
the our
conditions. including
and
the
the
s u i t a b l y m o d i f i e d s e t of equation
Navier-Stokes
interpolation space
A
the
of
equation
routine.
interpolation
Continuity,
would
should
Under probably
be
this be
1 28
w o r k i n g on a number o f 3D g r i d s r a t h e r interpolations differential
are
equations
T h i s w o u l d l e a d us modeling
just
the
like
set
of
the
it
i s my
belief
already
aspects
that
such
a l l
a
3D
framework, the
major
to
extension
above
and
resources.
challenge
some
atmospheric
special
in
sciences,
purpose
and
time
Because
routines
i n s t i t u t i o n s may
theoretical
p r e v i o u s l y and should
gave a s i m p l e
system.
With
the
a l s o be
also
computational
be
present
identified.
necessity must
to be
In choosing
follow
the
very
remaining
the
next much
part
of
trivial.
system
This
determine done
s o l v e d , the
o u t l i n e on a p o s s i b l e
more d e t a i l e d e v a l u a t i o n has
and
equations.
spend
t h e above o u t l i n e d p r o b l e m s a r e
p r o b l e m s can
definite
numerically
problem.
structure parametrization
to
of
data.
Rather than w r i t i n g such
I n t e r a c t i o n with other
same method d e s c r i b e d
The
governing
boundary
governing
p a r t o f c o h e r e n t s t r u c t u r e r e c o g n i t i o n can the
Such
from e x t e r n a l s o u r c e s .
rather
the
as
topic
worthwhile
routines
insight into
of the
If
is
one.
the a c t u a l
the
i n t e r e s t i n oceanic
exist.
g i v e us new
be
just
data
fluid
would
it
for similar similar
of
system t o 3D.
present
a routine ourselves, searching
initial
i n t o another heated
full
the
solving
w i t h the
Developing such r o u t i n e s extending
than
t o be
as
extension the
main
done b e f o r e a l l
investigation feasibility
before
committing
between u s i n g
the present
is
of
a the
substantial system to
do
1 29
more two
experiments, dimensional
o f a 3D s y s t e m ,
e.g.
t o study a subsurface
f l o w , and a c t i v e l y
engaging
layer
i n a near
i n t h e development
I w o u l d recommend t h e f i r s t o n e .
130
VIII. A computer
package
has
interpolation
and
turbulent
flow.
fluid
objective
is
further
turbulent The
been
developed
to
coherent
structure
recognition
to
from v i d e o used
to
extract
flow without
structures.
flow
resolution
readily
available
80%
of
flow
most
time
ordinary
the present
methods
in
i t s field
follow
picture.
the
the
a l s o being
the
volume
This
system
evolution
efficient for
of
of
enough
statistical
amount o f t i m e .
of view, the p l o t s of tracked
us w i t h no l e s s i n f o r m a t i o n flow
identified
objectivity,
information.
to
system,
improvement over
importantly,
structures while
a n a l y s i s w i t h i n a reasonable
provide
Using
s u f f i c i e n t coherent s t r u c t u r e data
Within
system
t o reproduce
a l l of the manually
quantitative
e n a b l e s us f o r t h e f i r s t
and
a n d a l s o g a i n new i n s i g h t i n t o t h e
turbulence.
and
2D
from t h e
preliminary
We have been a b l e
visualization
temporal
to provide
this
Such a system i s a d e f i n i t e
traditional
individual
using
one.
results
over
in
information
judgements.
established
of g r i d
This
s u b j e c t i v e manual
some
identified
field
quantitative
structures
results established
period
sufficient
coherent
i t t o be a p o w e r f u l
we
provide
flow v i s u a l i z a t i o n .
prove
initial
automate
I t was f i r s t a i m e d a s an e f f i c i e n t
method
information
CONCLUSION
Everything
t h a n we c a n afterwards
get
streaks from
an
i s a gain.
If
131
dynamical properties vorticity
are
structures, without
of
such as v e l o c i t y , primary
i n t e r e s t rather
t h e y c a n be e x t r a c t e d
much
energy,
difficulty.
from
Such
a
function of time.
the
system and c o u l d
This
the
tracked
studies allow
streaks
us t h e g l o b a l
parameter
w o u l d be a t r i v i a l
be u s e d t o f u r t h e r
and
than the coherent
c h a r a c t e r i s t i c s o f t h e flow and i t s v a r i o u s as
momentum
fields
extension
contrast
the
of
system
with established r e s u l t s . The part.
m a j o r work h e r e i s t h e c o h e r e n t s t r u c t u r e With
the
turbulent
flow,
valuable
tool.
growing
i n t e r e s t i n coherent
researchers With
should
this
package
a c q u i s i t i o n and a n a l y s i s system, establish
enough
understanding coherent This a
in
combined
structure recognition extension
recognition and
role
visualization,
is
addressed.
an
a
and
that
we
can
help
our
to
significance fluid
a
data
flows
of
the
i n general.
by m a j o r r e s e a r c h e r s " f o r
image p r o c e s s i n g
and coherent
system. outline
given
for
3D
coherent
structure
w i t h t h e major d i f f i c u l t i e s
The p r o b l e m s a r e by n o t t r i v i a l
not
away.
could
S i m i l a r systems with
indispensible for future
such extensions fluid
visualization
identified
b u t we
once we know t h e p r o b l e m , t h e s o l u t i o n w i l l
be
system
efficient
information
addresses the c a l l
flow
such
as
s t r u c t u r e , and thus t u r b u l e n t
particularly
An
exact
s t r u c t u r e s of
i t i s hoped
quantitative the
find
recognition
be
believe too f a r
t u r n out t o
studies.
132
BIBLIOGRAPHY A h l b o r n B., A h l b o r n F. a n d Loewen S. 1985, J . P h y s . D: A p p l . P h y s . 18, 2 1 2 7 . A h l b o r n F. 1902, "Uber den M e c h a n i s m u s d e s H y d r o d y n a m i s c h e n W i d e r s t a u d e s " , Abhandlungen aus deur G e b i e t d e r N a t u r w i s s e n S c h a f t e n , N a t u r s w i s s . V e r e i n Hamburg P u b l . L. F r i e d r i c h s e n & Co. A h l b o r n F. 1922, P h y s . Bareau
Z. 2 3 , 5 7 - 6 5 .
V. 1 9 8 5 , UBC P h y s i c s 459 p r o j e c t
report
C a n t w e l l , B. a n d C o l e s , D. 1983, J . F l u i d Mech., 136, 3 2 1 . C r i d d i e W.
1960, R h e o l o g y
Volume 3, A c a d e m i c P r e s s .
Crow S . J . a n d Campagne F.H. 1971, J . F l u i d Mech. 48, 547. Dewan e t a l 1985, UBC P h y s i c s 459 p r o j e c t H i n z e J.O. 1959, T u r b u l e n c e , M c G r a w - H i l l H u s s a i n A.K.M.F. 1983, P h y s .
report. Book Company.
F l u i d s 26, 237.
H u s s a i n A.K.M.F. 1985, "Forum on U n s t e a d y F l o w s S y s t e m s " , ASME.
in Biological
K l i n e e t a l 1967, J . F l u i d Mech. 3 0 , 7 4 1 . Loewen S. 1 9 8 3 , M a s t e r s
t h e s i s , UBC.
Loewen S., A h l b o r n B. a n d F i l u k A.B. 1986, t o be p u b l i s h e d P h y s . F l u i d s . A u g u s t 1986. P a v a n e t a l 1985, UBC P h y s i c s 459 p r o j e c t
report.
R o b e r t s o n J . A . a n d Crowe C T . 1975, E n g i n e e r i n g M e c h a n i c s , Houghton M i f f l i n . Sheu e t a l 1982, Chem. E n g . Commun. 17, 67.
Fluid
1 33
APPENDIX A - PARAMETER EXTRACTIONS FROM FITTED TRAJECTORY In the package, the streak p o l y n o m i a l f u n c t i o n s of t i m e : (A.1a)
(A.1b)
X(T)
Y(T)
= Z
A.T
= I
are
fitted
as
;
1
B T
coordinates
;
L
L
where the sum i s o v e r A. f r o m 0 t o some i n t e g e r K. From t h i s representation of X(T) and Y(T), we can derive the instantaneous parameters, most i m p o r t a n t l y the l i n e a r and angular velocities. Depending on the c o n d i t i o n of the experiment, the fitting models w i l l be different. Two s i t u a t i o n s are d i s c u s s e d here. First, when we expect the translational m o t i o n of t h e s t r u c t u r e t o be n e g l i g i b l e (as i n t h e c a s e of g r i d t u r b u l e n c e ) , we w i l l f i t (A.2)
V = wxR
;
i.e. assuming pure r o t a t i o n a l motion. This i s referred here as the s t a t i o n a r y model. A l t e r n a t i v e l y , f o r system or times where we e x p e c t t h e c o h e r e n t s t r u c t u r e s t r a n s l a t i o n be c o m p a r a b l e t o r o t a t i o n , we h a v e t o i n c l u d e t h e center mass m o t i o n . In such s i t u a t i o n s , the t r a j e c t o r y i s f i t t e d (A. 3)
V
=
Vcm
+
ZJXR
to at to of by
.
T h i s i s r e f e r r e d t o h e r e as t h e d r i f t i n g m o d e l . P a r a m e t e r s of interest are calculated from t h e s e f i t t i n g e q u a t i o n s . From t h e p o l y n o m i a l a p p r o x i m a t i o n of t r a j e c t o r y , theoretically we h a v e an i n f i n i t e number of e q u a t i o n s t h a t c a n be u s e d t o s o l v e any unknown p a r a m e t e r s i n any f i t t i n g e q u a t i o n . They a r e t h e v a r i o u s t i m e d e r i v a t i v e s : X ( T ) , Y ' ( T ) , X ' ' ( T ) , Y ' ' ( T ) up to any order we w a n t . However, t h e a c c u r a c y o f t h e d e r i v a t i v e s decreases with increasing order. This i s because of the u n c e r t a i n t y of t h e c o e f f i c i e n t s u s u a l l y i n c r e a s e w i t h o r d e r of the term. On d i f f e r e n t i a t i n g , t h e low o r d e r c o e f f i c i e n t s a r e successively e l i m i n a t e d , hence leaving the higher order derivative less accurate. So a s a g e n e r a l r u l e , we s h o u l d t r y t o m i n i m i z e t h e o r d e r of d i f f e r e n t i a t i o n s u s e d and a l s o t r y t o minimize the effect of t h e h i g h e r o r d e r d e r i v a t i v e s even i f they are invoked. 1
134
A.1
S t a t i o n a r y M o d e l : Vcm i s Z e r o In
t h e s t a t i o n a r y model, t h e b a s i c
(A.2) and
fitting
equation i s
V = wxR ;
a l l p a r a m e t e r s of i n t e r e s t s a r e d e f i n e d
(A.4a) (A.4b)
X ( T ) = Xc + R C O S ( C J T + 7 ) Y (T) = Yc + R s i n ( c o T + 7 )
i n the equations
; and .
(Xc,Yc) i s t h e center of r o t a t i o n , R i s the r a d i a l vector i n the CM frame, u i s t h e a n g u l a r v e l o c i t y and 7 i s t h e i n i t i a l p h a s e a n g l e a t T=0. T h e r e a r e f i v e unknowns, X c , Y c , R, CJ a n d 7. We r e q u i r e a t l e a s t f i v e e q u a t i o n s t o s o l v e them. As o u r equations always come i n pairs, we w o u l d be u s i n g s i x e q u a t i o n s up t o t h e a c c e l e r a t i o n t e r m s . D i f f e r e n t i a t i n g (A.2) w i t h r e s p e c t t o t i m e , a s s u m i n g t h e p a r a m e t e r s t o be constant, we h a v e (A.5)
A = uxV .
Consider the cross product a b o v e e q u a t i o n , we have VxA This
=
(A.6)
CJ
= o>V
2
VX(CJXV)
i s an e q u a t i o n
o f V a n d A a n d e x p a n d i n g A by t h e .
f o r CJ:
= (VxA) / V
2
.
—»
i
S i n c e CJ a n d R a r e p e r p e n d i c u l a r from e q u a t i o n (A.2) (A. 7)
R =
equation
we
obtain
( A . 4 ) , we have
Vx
= - R c j s i n (cjT+7) a n d
Vy
=
RCJCOS (cjT+7)
g i v e s u s an e q u a t i o n
(A. 8 )
system,
V/CJ .
Differentiating
This
i n o u r 2D
.
for7:
t a n ( c j T + 7 ) = -Vx/Vy .
By carefully considering the d i r e c t i o n of t h e v e l o c i t y c o m p o n e n t s , we c a n be s o l v e the i n i t i a l phase angle 7. F i n a l l y we c a n c a l c u l a t e t h e c e n t e r o f r o t a t i o n a s (A. 9a) (A.9b)
Xc = X - R C O S ( C J T + 7 ) Yc = Y - R s i n ( c j T + 7 )
= X - Vy/cj ; a n d = Y + VX/CJ .
1 35
A. 2 D r i f t i n g M o d e l : Vcm i s N o t Z e r o In (A.3)
t h i s model,
the basic f i t t i n g
equation i s
V = Vcm + u>xR ;
w i t h a l l parameters of i n t e r e s t s d e f i n e d i n t h e e q u a t i o n s (A.10a) (A. 10b)
X ( T ) = Xc + V x T + R s i n ( w T + 7 ) ; and Y ( T ) = Yc + V y T - R c o s ( u T + 7 ) . 0
0
The additional parameter (Vx ,Vy ) denotes the d r i f t i n g v e l o c i t y of t h e r o t a t i o n c e n t e r . As we introduce two more unknowns, we expect t o u s e up t o A', t h e t i m e d e r i v a t i v e o f a c c e l e r a t i o n A. D i f f e r e n t i a t i n g t h e a b o v e e q u a t i o n s o n c e , we have: 0
(A. 11a)
Vx(T)
(A.11b)
Vy(T) = V y
=
Vx
+
0
0
RWCOS(OJT+7)
+ Rwsin(wT+7)
0
;
and
.
This pair of equations i s very s i m i l a r t o equation (A.4). M o r e o v e r , i f we d i f f e r e n t i a t e t h e f i t t i n g equation once, we w o u l d have (A.12)
A = wxV
which i s a l s o very s i m i l a r to (A.2). Contrasting the previous model w i t h t h e new e q u a t i o n s , i t i s n o t h a r d t o i n f e r w i t h o u t anymore d e r i v a t i o n t h a t (A. 1 3) (A. 14)
(A.15)
u = ("AxA' ) / A Ru
tan(wT+7)
(A. 16a)
Vx
(A. 16b)
Vy
A/CJ
=
0
;
or
R
=
A/CJ
2
;
R C J C O S (coT+7)
=
Vx
-
Ay/w AX/CJ
= -Ax/Ay ; =
0
2
Vx
-
= Vy - R u s i n (coT+7)
= Vy +
; .
and
The o n l y r e m a i n i n g p r o b l e m i s t o s o l v e t h e c e n t e r o f rotation (Xc,Yc). This i s done by s u b s t i t u t i n g t h e p a r a m e t e r s b a c k into (A.10) and s i m p l i f i n g the r e s u l t i n g equations. We f i n a l l y get (A. 17a) (A.17b)
Xc = X - V x T + AX/OJ Yc = Y - V y T + Ay/w
2
0
0
2
; and .
1 36
APPENDIX B - USING THE PACKAGE AT
UBC
The package i s stored under t h e CCID "LKHA". T h e r e a r e 6 f i l e s p e r m i t t e d t o p u b l i c f o r t h o s e whose want t o t e s t r u n t h e s y s t e m . They a r e •
LKHA:RUN.LOG - a t e r m i n a l logfile containing sample runs of t h e v a r i o u s phases of t h e package. Comments d e s c r i b i n g t h e v a r i o u s s t a g e s are also included.
•
LKHA:O.LIB - L i b r a r y o b j e c t f i l e t h a t c o n t a i n s v a r i o u s r o u t i n e s r e q u i r e d by most p h a s e s o f t h e p a c k a g e . This file must be included f o r l i b r a r y search ( l i n k e d ) before running the package.
•
LKHA:0.CNTR - O b j e c t f i l e f o r t h e n o i s e t r a c e r c e n t e r i n g phase.
•
LKHA:0.STK phase.
•
LKHA:0.ANA - O b j e c t f i l e f o r t h e f i e l d interpolation and r e c o g n i t i o n p h a s e . The p a r a m e t r i z a t i o n p h a s e is includes a s a s u b r o u t i n e i n t h i s f i l e a s we merged t h e r e c o g n i t i o n a n d p a r a m e t r i z a t i o n part in the package. U s e r s can s e l e c t whether they want t h e p a r a m e t r i z a t i o n a n a l y s i s d u r i n g t h e r u n .
•
LKHA:PRIMARY - S a m p l e d a t a f i l e t h a t i s t r a n s f e r r e d t o MTS f r o m t h e m i c r o c o m p u t e r by AMIE. T h i s i s t h e i n p u t f i l e f o r LKHA:0.CNTR.
-
Object
file
reduction
f o r the streak
and
connection
Moreover, several system routines must a l s o be l i n k e d b e f o r e running the package, these i n c l u d e the g r a p h i c s package * I G , t h e IMSL d o u b l e p r e c i s i o n l i b r a r y IMSL:0.9D a n d a l s o t h e main l i b r a r y *LIBRARY ( u s u a l l y l i n k e d a u t o m a t i c a l l y by MTS). For r e a d e r s i n t e r e s t e d i n the s o u r c e code of t h e package, they should refer to Professor Boye A h l b o r n of the Physics Department. The c o s t s of running t h e v a r i o u s p a r t s of the p a c k a g e a n d t h e i r 10 a s s i g n m e n t s c a n be f o u n d i n t h e l o g f i l e .
