ý-
OF
UNIVERSITY
-/
LONDON
FLOW ALONG
LAMINAR
CORNERS HAVING
ARBITRARILY
LARGE TRANSVERSE CURVATURE
by
ADIL
HADI
B. Sc.
A Thesis
of
Philosophy
in
in
the
(Eng.
for
submitted
RIDHA
the
)
Degree
Faculty
the
University
of
of
of
Engineering
London
1978
Department
of
Mechanical
College
University
u'v
Engineering
London
Doctor
---
CD c> i
-i
COPY
BEST
AVAILABLE
V
00
ri
qI
ble
print
ity
2
ABSTRACT
Theoretical
of
flows
to
corners
The
in
and
new
work
region
intersecting
by
is
finite
I
curvature
which
(radiused
corner).
The
invariaýt
with
of
A
thickness.
is
motion
corners
the
to
sharp
corners
of
Numerical
included
sharp
angle
and
from
Results
a rectangular
for
solution
for
obtained
from
the
ment
although
theoretical
at
corner
flow
are
point
velocity
show
within
is
the
somewhat
for
between
an
and
corner
The
of
the
only
angles
shown.
given
fair
of
results
corners
is
applicable
corners
excellent..
measurements
results
any
as
is
180
agreement
sharp
such
contains
and
case
0
90
sharp
inexact
are
along
include
is
equations
It
for
here
the
layer
the
problem
obtained
the
given
available
from
of
than
The
be
and
boundary
less
0
315
and
corners.
radjused
with
experimental
are
0
45
to
0
90
the
the
or
to
chosen
coordinate
a special
greater
from
is
surface
planes
generality.
as
solutions
solutions
different
handle
the
where
the
formulation
geometry
corners).
a distance
as
confined
symmetrical
streamwise
considerable
angles
(sharp
a
surface
order
to
results
corresponding
other
of
radiused
exact
existing
joins
within
same
deveaoped
corner
smoothly
analytical
new
and*is
by
the
plate
the
of
plane
symmetry
of
a flat
to
asymptotes
to
been
situations
replaced
curved
respect
hitherto
concerns
behaviour
the
of
planes
here
presented
intersection
of
have
corners
streamwise
formed
studies
experimental
of
the
and
flow
when
compared-
qualitative
boundary
higher
along
agreelayer
than
the
predicted.
3
ACKNOWLEDGMENTS
The
in
Department
the
College
criticisms,
and
His
valuable
and
above
the
friendly
of
great
help
lecturer
University
investigated
in
constructive
sound
his
all,
Barclay,
problem
discussions,
were
encouragement
H.
Engineering,
suggested
who
W.
Dr.
to
Mechanical
of
London,
text.
this
indebted
Is
author
attitude,
in
patience
this
making
work
possible.
the
The
assistance
design
and
the
of
making
Departmental
the
of
corner
Workshop
is
model
in
staff
gratefully
appreciated.
The
Ministry
would
author
of
The
Oil
help
his
friends
and
enjoyable.
been
and
and
Finally,
Ms Ketty
without
have
not
Shoet
to
wishes
whose
for
to
thanks
financial
the
Iraqi
this
support
research
concluded.
encouragement
colleagues
the
his
express
author
typing
to
extended
made
renders
the
this
his
thesis.
work
sincere
the
more
by
author
interesting
thanks
to
4
CONTENTS
PAGE
NOTATION
6
0
CHAPTER
INTRODUCTION
2
1.1
General
1.2
Review
11
remarks
of
THEO RETICAL
on
the
literature
some
flow
problem
11
corner
flow
15
corner
on
ANALYSIS
2.1
Introduction
2.2
Choice
2.3
Boundary
2.4
Flow
39
39
of
coordinate
layer
along
a
2.4.1
Corner
2.4.2
Equations
40
system
41
equations
corner
50
geometry
of
50
motion
in
zero
pressure
51
gradient
2.4.3.
3
2.5
Asymptotic
2.6
Remarks,
2
Method
.7
the
on
of
3.2
Corner
model
3.2.1
Corner
3.2.2
Model
Preliminary
4.2
Sharp
asymptotic
crossflow
64
crossflow
velocity
w 68
71
78
78
Experimental
4.1
the
WORK
Introduction
AND
of
52
solution
3.1
RESULTS
conditions
decay
EXPE RIMENTAL
3.3
4
Boundary
79
79
profile
design
arrangement
DISCUSSIONS
82
and
measurements.
89
107
107
corners
112
5
1
4.3
4.4
4.5
5
4.2.1
Streamwise
4.2.2
Crossflow
4.2.3
Wall
Radiused
velocity
velocities
shear
112
v
and
w
stress
126
corners
4.3.1
Streamwise
4.3.2
Crossflow
4.3.3
Wall
127
128
velocity
velocities
shear
Sharp-profiled
v
and
w
Streamwise
4.4.2
Crossflow
4.4.3
Wall
136
Experimental
137
velocity
resu
137a
field
velocity
shear
132
135
stress
co rner
4.4.1
119
138
stress
139
lts
CONC LUSIONS
246
APPENDIX
A
On the
boundary
asymptotic
by Desai
16
and Mangler
B
Solution
C
Numerical
REFERENCES
of
of
equations
derived
249
(2.49)
equation
solution
conditions
255
(2.66)
257
264
6
NOTATION
Important
All
A,
symbols
are
defined
A
in
used
symbols
where
a
A
(i
A(i)
physical
n pB n
a,
text
they
first
are
below.
given
appear.
constants
A
A
the
Cn
(dummy)
= 1#2#3)
see
b
vector
contra-varient
A
of
component
(2.41)
equations
in
constants
component
,
transformation
the
of
into
B,
C
B
constants
wall
r
U
= R_'
er
r
c
Cos
c
Cos
D,
F*I
curvature
D
E,
E
C
I
see
(2.66)
equation
d
diameter
of
d
a length
geometry,
scale
characterizing
(4.5)
equation
f
Blasius
gij
metric
H(j
a function
velocity
h
step-size
formulation
h
Coefficient
i
Jacobian
equation
k
thermal
hot-wire
function,
(2.45)
equation
tensors,
(2.4)
equation
to
related
*
w,
equation
in
corner
the
of
finite
heat
of coordinate
(2.5)
conductivity
crossflow
(2-62d)
difference
transfer
transformation,
of
A
7
L
length
,e
scales
M
number
of
direction
formulation
m
exponent
equation
N
number
direction
N
p
pressure,
p
pressure
'POI
1511 P2'
R'Q
L
in
the
(2.17a)
equation
of
0
in
p
kJ*-/,.,,
Reynolds
Reynolds
number
local
Reynolds
number
curvature
r/
radius
of
curvature
s
sin
s
sin
s
defined
at
d
distance
Lila
free-strea
gradient
1
U( x
p.
equati
t
from
at
symmetry
plane
65
on
(4. '6)
symmetry
m velocity
plane
in
in
mm
zero-pressure
U10 _jM
u , Vr W
0
velo
c iti
U"
/10
es
i n di rect
i ons
V
voltage
across
the
hot-wire
V
voltage
air
across
the
hot-wire
0
v 1,
e
number
local
,
of
v
of
expansion
(2.14)
2LJxl/r..,,
r
1
variation,
number
radius
U
stream
planes
Nusselt
Up* xIA,,,,
ex
r0
free
the
difference
eu,..IL,
x
R
mesh
coefficients
equation
Re
R
of
h.0 d/k,
u
in
mesh planes
finite
in the
for
the
(2.16)
43
23
vv
v2,
contra-variant
vector
V3
co-variant
componen.
components
ts
of
in
of
still
velocity
velocity
vector
8
v(2),
v(l),
vw
v(3)
physical
vector,
of
(2.12)
components
equation
in
crossflow'velocities
cartesian
coordinates
1
-2
vV
v 21
-3
velocity
non-dimensional
(2.14)
equation
velocities,
v3
non-dimensional
(2.14)
variables,
x1,
x2x3
non-orthogonal
-1
x
-2
xx
y
1
23
yy
Or.
13
-3
rectangular
A
coordinates
to
related
1f/
(1;
coordinates
coordinates,
cartesian
quantities
A
equations
curvilinear
non-dimensional
(2.14)
equation
lim
rectangular
u,
v
and
w
_f)
coo
two-dimensional
boundary
defined
at
p.
40
defined
at
P.
50
layer
thickness
constant
a
dummy
(boundary
layer)
coordinate
viscosity
kinematic
P
viscosity
density
curvilinear
(2.17a)
coordinates,
variables
(2.17a)
equation
to
related
v.
u,
w.
equation
C
non-dimensional
in
the
system
(i
=
streamwise
3),
1,28
variables,
a quantity
vorticity,
rel ated
equations
equation
to
vorticity
coordinate
(2.33)
streamwise
(2.17a,
c)
C
COM
TxC
physical
in
direction
component
x'
R2), dehned'at
of
the
p. 108
vo .rticity
9
independenz
(2.64)
transformed
equation
YA
OL
=
VWC
maximum
scheme
value
of
maximum
scheme
value
of
variable,
in
the
in
numerical
the
numerical
layer
shear
stress
in
th e corner
shear
wall
region
stress
in
th e two-dimensional
wall
2 sin A
-.
zS'
COS2
Cr
)
ý
C;
Cr Sin
To
0ý sY.
S
VC.
Cos;k 6 X
ýX Z,
C
5Z2 j)ýý 3'
Cos "
C
+z
1
11
COSN
31
Dz-2S;
00
06
Aý
V% 42 _ý_
C)Lil
4e
C
.
Cos
+
Co's A
(ae7
Iv
=1[JA
zý
4z
c)
2L
L?
Z. - ;t 42,5'.', A
+Z
b2
k so'.
%'XýL
+ Co%
10
SuDerscriDts
i,
J,
k
denote
the
of a vector
etc.
a prime
respect
component
contra-variant
denotes
to
differentiation
denote
non-dimentional
with
variables
used
system
in
denotes
coefficients
large
F., when used
of
with
with
quantities
the
x
when
coordinate
for
expansions
v. and w
Subscripts
1,2,3,..
i,
J,
etc.
k
etc.
refers
used
to
with
indicate
vector
order
flow
of expansions
variables
co-variant
Pi
denotes
respect
parttal
to
ý&
;i
denotes
the
xi
co-variant
co-ordinate.
component
differentiation
derivatives
when
of
a
with
in
11
CHAPTER 1
INTRODUCTION
1.1
by
General'
joining
two
quarter-infinite
has
attracted
edges
the
case
In
the
line
line
free
undisturbed
there
exists
from
arising
the
the
This
in
provided
a valuable
of
in
the
which
The
the
join
sharp
of
the
In
in
replaced
')
infinite
corner
not
is
e
against
or
its
greater.
curvature
above
the
by
a joining
as
the
example
new
work.
in
region
the
but
infinite
This
has
treat
one
that
if
region
classic
to
present
sense
corner
layers
treatment
geometry,
require
the
flow
which
the
of
the
parallel
of.
the
of
and
and
boundary
the
three-dimensional
does
is
vicinity
one
now
fluids,
subject
laterg
O(R
the
Fig.
coplaner
line')
of
is
of
the
(see
three-dimensional
problem
apparent
line
corner
in'cludes
of
the
become
curvature
the
presence
mathematically
will
stream.
background
be
will
of
vicinity
('corner
of
particularly
are
edges
their
at
plates
corner,
corner
category
formed
corners
attentionj
leading
mechanics
a wider
flat
interaction
mutual
plates.
thin
inherently
an
problems
flow
the
of
along
right-angle
the
case
problem
flow
considerable
internal
an
every
joining
the
on
of
flow
'corne'r
layer
laminar
(1.1)).
to
boundary
The
side
in
*6rf 'tlie'
'fe'ma:rks'
will
as
exist,
surface
a special
be
corner
curvature
condition
the
of
of
case.
at
maximum
course
12
(a)
(1.1).
Flow
Flow
Fig.
(a)
(b)
Apart
problem
of
Corner
along
an
along
an
from
its
flow
along
since
flow
systems.
a corner
other
hand
are
blade
the
The,
of
incompressible
wise
to
flow
corners
of
but
with
angles,
common
spaces
problem
having
applicable
examples
of
which
finite
curvature
of
turbomachinery.
be
treated
along
no
is
found
are
in
in
this
geometrically
a wide
shown
in
important
on
is
variety
Fig.
laninar
the
streamand
(1.2).
corner
It
arbitrary
The
variation.
of
the
notably
curvatures
curvaturý
to
in
symmetrical
transverse
streamwise
curva-
engineering,
work
the
practical
rarely
occurrence
arbitrary
analysis
a
transverse
immediate
more
are
to
a solution
finite
of
of
corners
Corners
configuration.
corner.
corner.
generality,
is
'sharp'
value
flow
internal
external
greater
(Iradiused''corner)
ture
in
(b)
is
shapes$
im portantp
13
-I
4J
sJ
(a)
F,
(b)
..
ý
(0)
I
Pig-(1-2)
Corner geometries
(a) radiused
(b) shaýp
corner,
(c) sharp-profiled
corner,
corner.
14
to
however,
which
asymptote
O(R
from
e
One
an
to
show
by
stable
it
separate
adverse
is
found
in
the
a finite
stress
flow.
the
The
and
incompressible
The
with
Velocity
traverses
results.
with
to
is
the
be
profile
and
anemometer
(at
the
devblopment
compared
be
with
small
the
analytical
flow
along
(internal
and
is
mainly
a rectangular
symmetry
is
the
line)
calculated
corresponding
the
is
in
effect
the
stability
work.
0f
two
parts;
deals
part
with
radiused
and
external
corners).
ihat
radiused
of
a
a non-
produce
present
The
flow
the
and
to
consists
layer
will
arbitrarily
here
programme
experimental
curvature
in
experimental.
angle
line
of
sharp
Ill_
improvement
consequent
the
a marginally
corner
curvature
therefore
presented
in
only
behaviour
infinite
stress
two-dimensional
stress
the
a seemingly
This
shear
experimentally
near
confirmed
arbitrary
a hot-wire
maximum
of
boundary
of
flow
the
out
easy
shear
in
be'in
may
is
wall
wall
shear
corner
It
separation
at
borne
will
This
work
analytical
corners
of
Zero
flow
gradient.
curvature
shear
zero
flow
is
presence
consequence
of
radiused
the
corner
line.
the
that
shapes
flow.
the
of
vanishing
This
pressure
direct
the
the
of
sharp
that
condition.
where
stability
inference
a warning
to
a distance
within
advantages
corner
of
limited
symmetry.
the
the
at
a characteristic
is
of
internal
an
vanishes
corner
surfaces
plane
in
is
analysis
anticipated
for
and
the
plane
the
of
problems
of
the
that
stress
of
to
improvement
is
is
that
note
0.1
from
of
sharp
traversing
corner
mM-
10
these
theoretical
of
15
1.2
of
To
author's
the
flow
corner
on
detailed
is
literature
[3]
Barclay
to
prior
the
it
problem
dimensionality
equation
of
O(R
ted
first
one
deals
review
present
individual
which
requires
each
coordinate
of
The
direction.
only
extends
to
Past
symmetry.
these
solve
involvement
the
of
solutions,
to
three-
inherent
the
of
with
re-
need
attempts
region
both
[2],
Zamir
mainly
layer
boundary
plane
parts
work.
is
or
by
notably
first
the
incorrect
attempted
Loitsianskii
15]
sharp
by
corner
which
who
characteristics
some
was
than
0.
180
the
considered
an
well
of
which
to
have
are
with
to
Both
works
made
the
deal
the
momentum
treatment
Bolshakov,
with
failed
to
been
by
a rectangular
of
the
of
in
established
work
case
adaptation
Later,
his
generalized
appears
solution
using
flows.
dimensional
integral
of
two-
Loitsianiskii
coriers
of
satisfy
the
angles
conditions
above.
stated
[7]
Carrier
boundary
the
the
different
to
below.
The
less
Somewhat
at
from
e
in
corners.
dimensional
three
recognize
discussed
method
for
the
to
resul.
motion
that
distance
failure
The
of
is
second
poses.
flow
sharp
corner
looking
the
available
This
the
to
of
literature
workers
relevance
of
the
accorded
previous
of
emphasizing
[6]
been
[4].
properties
has
have
El-Gamal
Two
a
to
and
publications
the
confined
by
flow
on corner
knowledge,
treatments
the
of
some literature
Review
case
layer
of
equations
0
a 90
sharp
first
for
corner.
attempt
the
corner
He
used
to
satisfy
flow
only
the
when
considering
the
streamwise
16
the
concerning
the
continuity
and
momentum
velocity
relationship
but
expectation
crossflow
equations
sharp
of
four
by
continuity
flow
and
only
an
to
yield
in
(9)
[10]
Dowdell's
answered
Carrier's
used
all
the
conditions.
týat
Dowdell
from
the
satisfied
only
Pal
order
the
the
pressure
the
the
cross-
resulting
the
correct
900
a
Carrier's
corner
solution
inexactt
though
the
near
where,
El-Gamal
[4)
omitted,
equation
continuity
and
was
motion
on
error.
we have
and
as
the
and
left
un-
of
Dowdell's
Dowdell
boundary
correct
this
by
a result
Rubin
noted,
considering
presumably
since
consequently
coincidence
as
has
Grossmang
and
in
grossly
of
the
fourth
satisfying
for
Carrier'st
that
equations
vorticity
gradient.
two
to
the
results.
were
had
to
internal
an
to
from
with
solutions
pressure
approach,
association
for
zero
results
satisfactory
reason
of
subject
latterts
[11],
case
scheme
different
solutions
the
omitted
all
eliminating
iterative
Dowdell's
the
satisfying
subtraction"of
numerically
slightly
Rubin,
shown,
and
and
solved
consequently
the
of
functions
stream
identically
Using
ver*y
physical
attempted
reduced
were
two
conditions.
assumed
and
introducing
were
boundary
were
motion
with
cross-differentiation
equations
importance
the
angle
of
equations.
of
with
consistent
a solution
treated
arbitrary
equation
by
terms
He
equations
equations
assumption
components
supervision,
finding
by
motion.
corner
the
Carrier's
under
matter
of
three
was
of
an
remained.
[8],
the
resolve
solution
question
equations
Dowdell
The
the
the
amongst
The
vector.
by
augmented
equations
two
oversight,
of
streamwise
found
point,
which
his
momentum
terms
solution
17
In
Carrier's
first
his
distinct
in
the
O(R
to
apply
from
the
but
through
the
region
IV,
'corner
the
three-dimensional
Having
of
of
manner
of
fl, ow
problem
similar
to
an
in
0
90
a
a
sharp
earlier
side-edge
clarified
the
complete
the
solution.
the
set
method
formulation
systematic
This
corner.
of
essenýially
the
used
was
in
by
Stewartson
(12]
a quarter
infinite
flat
treatment
the
near
Rubin
expansion
for
for
required
distinction,
this
asymptotic
flow
the
is
of
short
is
flow
the
is
there
Finally
which
nothing
equations
made
matched
the
here
arising
boundary
region.
in.
regions
equations
effects
respective
flow
layer',
and
layer
the
In
layer
dimensional
of
represents
ei).
boundary
three
potential
I
O(R
to
the
four
by
(1.3a).
Fig.
potential
interaction
mutual
boundary
is
with
layers
of
in
that
represented
two-dimensional
usual
e-
demonstrated
be
can
flow
the
i)
Rubin
depicted
as
where
III
and
[9],
a corner
regions"
region
method.
paper
layer
"boundary
the
in
as
equations
plate.
Rubin
the
between
the
in
has
correct
layers
boundary
conditions
He
found
f
tiation
constitutes
is
with
regions
for
the
1r
p
function,
Blasius
the
to
respect
one
of
n
and
the
II
and'IV
the
c).o the
to
wall
and
flow
variables
low
crossf
y=0
derived
is
given
by
1
ý6_r---T7O;
a prime
13
interaction
of
all
as
!12
q LLx )
kr
where
that
w parallel
component
velocity
in
boundary
IV.
region
mechanism
denotes
differen-
(7f
This
7-0.00
the
boundary
conditions
on
region
IV
as
18'
X> Oly > of
z0
x
>02ýY>-Olz
>0 ID,z>0
IN
:,. >O#Y>O1z>-
0x
>01Y>Osi>0
(a)
Z,
Outflow
regi
,
frox
gn III
Resultant
P. 0-
Acq
0<ý<C?
jP
)---ýO
UI
.0w
-rl
frox
Outflow
O
egi on
C-4
VIV
profile
of Cr*ossflovt
component.
w
IV
--br.
(U/2,2x)
z
Fig-(1-3)
Corner
flow
geometryo'
A0
19
in
but
IV
boundary.
the
tj C t4
t7
1
;::r, .5ý
the
that
it
that
decay
Rubin
analysis,
solution
for
remarked
above
was
shows
their
in
the
symmetry
layer
in
showed
error.
This
for
found
that
streamwise
of
asymptotic
flow
all
(regions
layer
vorticity
and
Using
region.
a numerical
As
domain.
Carrier's
be
can
and
that
produced
a bounded
results
results
the
flow
[111
III
values
velocity
potential
exponen-
and
boundary
the
Grossman
and
corner
substantially
which
the
the
and
streamwise
the
II
investigated
region
of
values
decay
finite
at
region
indicated
not
regions
into
into
exponentially
this
then
the
only
did
(1.1)
apply
layer
small
computations
into
algebraically
while
and
variables
[10]
for
necessary
finite
at
Preliminary
to
corner
decay
set
algebraically
the
variables
solution
flow
and
of
III)
be
.
Rubin
behaviour
and
to
incorrect
Pal
II
had
layer
only
was
n*umerical
47,
ZX r Say)
corner
but
tially.
the
solution
in
seen
velocity
(1.4)
Fig.
in
profiles
plane.
i
Although
rigorous
first
11
regions
to
and
date
find
the
IV
(and,
establi'shed
by
Pearson
the
to
arrive
as
problem
equation
decay,
numerically
(1.1).
however,
his
and
to
analysis
the
not
Rubin
for
incorrect
the
and
IV).
[131
who
utilized
this
an
resulted
(see
Fig.
to
in
(1.4)).
most
he
was
for
requirements
These
equation
failed
problem#
matching
III
at
the
produced
corner
correct
Pearson
and
colleagues
were
earlier
the
symmetry
essentially
the
recognize
a solution
the
which
of
same
algebraic
was
20
Rubin
Grossman[111
Pearson[131
Carrier [71
.0
I.
An
by-the
alternative
terms
is
the
of
due
boundary
and
plane.
problem
crossflow
equations
in
them
transformed
symmetry
treating-the
of
decaying
Rubin's
with
in
velocity
method
algebraically
Starting
N
Streamwise
(1.4).
Fig.
-8
-6
.4
1-
posed
he
conditions
independent
new
[14).
Ghia
to
variables
S where
and
NS
C
and
C is
bounded
onto
by
the
finite
the
which
were
transformed
symmetry
numerically'by
throughout
remove
this
or
N, S-,*-'l
quarter
and
the
becomes
conditions
direction
.
domain
infinite
wall
Dependent
inconvenience.
alternating
0
the
4,1
boundary
their
an
way
O,< N, t
as
unbounded
'with
this
plane7=4
region.
to
equations
scheme
In
a constant.
large
The
were
implicit
7=0
is
mapped
variables
were
new
then
readily
set
solved
(ADI)
of
21
The-crossflow
well
Rubin
with
by
differ
results
velocity
results
in
100,lo'near
the
Grossman's
and
a maximum
of
about
found
were.
to
compare
value
peak
but
trend
general
atl=ý-0-4.
'Cv(NN)
0.0
$0
0.4
0.8
2.8
2A
1-6,2.0
1.2
6h
LaD
41
PRES
WNT
.-(
0 RUUN AND (101
aG
N 0.6
04
02
;ý010
00
0.4
0.2
06
0.0
10
12
D(N,'N)
Velocities along symmetry line of infinite corner.
(1.5)
Fig.
velocity
(1.5)
Fig.
flow
and
In
velocity
streamNvise
ý
for
seems
boundary
convential
this
the
coordinate
di, fference
this
in
the
velocities.
the
symmetry
plane
given
corresponding
Ile
is
reported
in
still
made
Ghials
differences
that
when
plane.
between
agreement
Professor
request
his
of
numerical
Grossman's
and
in
re-plotted
in
is
solutions
are
are
cross-
Ghials
from
symmetry
Rubin
coordinate
agreement
two
u
a copy
with
the
difference
explaining
writer
the
shows
figure
On
together
layer
and
the
that
excellent.
present
1151,
in
for
in
crossflow
[14]).
velocity
solution
used
the
Thpse
results
flow
the
for
results,
of
;
kindly'sent
Ghia
for
reference
and
u and
(reference
[141
from
coordinate
solution
velocity
plane
taken
Grossman's
and
(N, S)
the
the
In
w
is
Rubin
Streamwise
in
symmetry
(1.6).
Fig.
fair
but
the
The
clearer.
in
paper
in
solýing
the
the
the
the
reasons
course
crossproblem
22 -
0.0
Rub'*in Grossman[lll
Ghia[151
-Desai
0.1
& Mangle.
r[161
1111
-j
u
Fig. '(1.6)
Streamwise
velocity
symmetry plane..
in
23
in
a finite
Pal
region
influenced
cýo
the
remarks
for
comparison
of
a rectangular
The
[161
Their
angle.
in
component
They
analysed
who
the
they
limitation
a general
is
The
problem
from
taken
as
and
the
reference
in
results
the
case
in
in
system
but
boundary
the
to
in
shown
outset
system,
the
the
the
of
and
arbit-
velocity
(1.6).
Fig.
this
within
layer
III
(ý
0ý
equations
1ý
).
direction
streamwise
planes
and
constant.
boundary
usual
chosen
Desai
a corner
streamwise
is
the
by
with
for
coordinate
parallel
particular
solution,
dealt
the
expressed
lie
coordinates
3
stretched
are
and
the
plane
curvilinear
Coor'dinate
31
been
for
symmetry
have
conditions
of
author's
also
solution
similarity
assumed
is
should
corner.
has
coxner
it
boundary
solution
the
of
solution,
method
present
sharp
sharp
Mangler
the
with
was
because
asymptotic
Ghials
his
above,
made
the
by
at
arrived
solution
that
of
the
of
view
the
concluded
place
In
series
that
behaviour
to
accurate
at
in
.
asymptotic
found
was
Ghia
asymptotic
more
rary
it
bytmaz
algebraic
be
(101,
Rubin
and
the
using
for
the
layer
corner
sense.
is
problem
coS 13
[(COS
and
is
13 + ýA-sin
X
shown
Cartesian
edge.
2
U..
in
system
and
V
Fig.
+4
(1.7),
with
ar'e
origin
the
free
where
tatT
123
(x .xx)
the
onýsymmetry
stream
velocity
form
plane
at
and
a rectangular
the
leading
kinematic
24
Image removed due to third party copyright
Fig.
(1-7)
Sketch
of the general co-ordinate
From -Desai and Mangler 1161 .
system
Image removed due to third party copyright
Pig. (1.8)
Flow in the cross plane.
From Desai and Mangler. ý 16 1-
25
viscosity
respectively.
Desai
to
similar
from
the
Mangler
the
model
by
paper
The
from
and
that
for
Desai
the
Rubin
the
t. o
was
and
the
is
wall
i.
zero,
subdivision.
different
[11)
and
vector
at
is
flow
the
[14]
Ghia
1
component
crossflow
e.
taken
significantly
velocity
solution
is
this
show
Grossman
crossflow
Mangler's
and
parallel
0
a 90
corner
by
obtained
to
regions
(1.8)
Fig.
Mangler
and
for
solution
particularly
In
Desai
Rubin.
by
used
into
the*flow
subdivided
strictly
2
two
dimensional
in
Pearson's
[131
with
The
2
vanishing
They
x2
Ix 21
to
large
(where
This
is
and
is
the
on
a secondary
by
and
indeed
the
is
so
This
ý
point
The
2
0.0
this
vicinity
from
This
results
component
course.
outflow
-oa,
which
velocity
when
In
the
physical
of
applied
as
wall
Mangler.
the
1
ý--.
(1.1).
equation
to
to
at
wall
layer
shear
Desai
small.
subjected
the
to
not
(1.1).
equation
parallel
be
arbitrarily
layer
layer
to
contrasts
results
corresponded
vanish).
fixed
distribution
the
rise
gives
in
the
is
clarified
velocity
in
(1.3b).
Fig.
A related
for
assumption
condition
boundary
boundary
an
being
[111
component
00
matching
with
the
to
would
question
proper
and
This
00
Rubin's
that
assumed
coo'rdinate
in
region
crossflow
amounts
-Oo
the
the
each
900
case,
factor
contributory
corner
on
the
is
the
crossflow
region.
As
the
potential
parallel
to
the
symmetry
velocity
vector
attains
different
is
region
a'constafit
the
in
vector
the
in
approached
the
of
magnitude
value
results
imposed
condition.
velocity
plane
the
affecting
to
O(R
e
in
potential
a direction
crossflOw
which
is
26
to
analogous
This
plate.
Desai
is
the
is
crossflow
velocity
potential
region
and
be
parallel
vector
must
In
consequence
by
the
of
from.
used
change
the
the
of
the
crossflow
the
of
the
flow
at
weaker
condition
vector
with
(1.3b)).
determined
be
may
z
ý
vector
the
of
(Fig.
plane
layer
vorticity
direction
the
symmetry
and
throughout
same
velocity
only
the
magnitude
that
symmetry
boundary
the
the
Rubin's
since
the
be
must
crossflow
Mangler
and
that
to
to
solution
Desai
rate
vector
the
Pearson's,
to
flat
semi-infinite
However,
follows
it
over-a
common
solutions.
here,
zero
flow
of
condition
Mangler's
and
also
situation
.
_____v_oo
that
the
to
respect
I
was
zero
The
at
are
solution
These
and
described
in
some
defects
in
an
are
have
Mangler
general
curvilinear
be
to
used
treat
Zamir
in
a general
to
prior
system
are
equations
To
analysis.
analysis
in
Zamir's
An
at
of
some
the
alternative
this
result
a
may
equations
problems
but
these
coordinate
flow
some
is
necess.
These
problem.
in
defects
ary
for
not
unfortunately
of
it
boundary
the
curvilinear
corner
utility
to
the
the
the
consider
detail.
derivation
representation
for
equations
consequence
clarify
the
and
three-dimensional
in
in
equations
three-dimensional
of
derived
some
Desai
work.
assumed,
analysing
intended
purpose
tensor
[171
layer
the
(A).
Appendix
valuable
system
variety
be
can
in
boundary
the
a wide
[21
layer
derived
detail
for
this
of
otherwise
coordinate
similarity
where
consequences
from
commences
of
energy
but
equivalent
in Chapter
per
2j
unit
consideration
volume
is
approach
55-56.
pp.
used
of
of
the
a fluid
to
arrive
27
in
element
motion.
If
10Ll
IaIpu
T"
tz
,zr
uIll
pui
2f
a
IOU
12
2
pd U-v
1
PU
zx
wherep
denotes
varient
component
33ýp
10
uI
p
density,
V the
the
of
PU2
pressure
velocity
and
is
uk
the
contra-
vector,
J4
U
is
where
14
a general
curvilinear
is
t
and
f
dt
a time
a comma
before
differentiation
Zamir
of
the
the
in
't)
indicates
coefficient
terms
following
the
velocity
is
that
assumed
91,
having
e
covarient
of
the
of
viscosity.
physical
componente
hold
luf
(1.4)
U'u,
specified
as
6 U,
6 S,
t.
(Ilfk
a subscript
and
system
coordinate
7=/1(q
where
coordinate
the
streamwise
ýUi
(i=1,203),
:: ý, as!
and
coordinate,
(1.5)
(J=1,3)
0'
J.
where
dS
is
Assumptions
without
to
the
(1.4>
an
order
energy
and
of
tensor
the
(1.5)
magnitude
(1.3).
line
element
were
directly
comparison
This
resulted
along
applied
with
in
other
(i.
e.
terms)
28
0,
C9p
22ýU'
Ulu,glip
FUZ
u2
p tt3
to
the
represent
Incompressible
coordinate
curvilinear
the
and
C
ij
C
and
tau,
.
4j
13j(. au
++
+0
when
loped,
(1.7)
direction.
9j,
and
vanish.
equations
were
solid
was
from
to
agilul 1
942
a 91,
L 922ag
21
+.
F47 I
Lg-u
+
+
ae)
-
in
the
lead
continuity
2.
/UAO
Equations
these
The
?
4"
v=
changes
orthogonal
"If
04J
where
an
7' +
L
tl
flow.
obtained
which
1apa21
1)., ql
then
agt,
layer
and
adopted.
were
5-2 +
ae
-L oil
was
(J=1,2,3,4)
=0
u3w
Ai
system
a boundary
considered
was
equations
streamwise
=0
,ýtls
flow
steady
in
tensor
energy
highly
(1.8)
This
applicable
surface,
curved
over
or
were
test
tested
was
to
which
for
to
t aken
cases
where
verify
that
three-dimensional
the
discontinuous
layer
boundary
in
problems
the
lateral
deve-
29
following
The
describe
coordinate
a sharp
system
right-angled
i
z
4 =x
Yz
3594=
4
;,
.
(x,
where
with
the
axis
x
On substituting
uu+
ae
.
(Y z
3)
4-13
i
'y 4 -4- Zf
in
a cartesian
along
the
corner
line
as
(1.9)
into
(1.7)
and
(1.8)
reference
frame
in
(1.9).
shown
the
Fig.
following
obtained
were
equations
y 4 *, Z 4
coordinates
are
y, z)
3)1/3
(Y'
Z)
ý
to
corner
94 ýý VT
ýz
911
introduced
was
(Y 4.4.Z 4)
Cy * 2) 2
4+
y
Z4-)
()fj - ZI)2/3
tL
d42
W
a 43
((Y+Z)CY4+z+)
zYz
to
-14
a4
+
(Y+Z)""
32
+2z(Y
141
du
dV
(y*. fz9
-S-&i + (.Y+Z)g 04"
+
p4,,
-IF
T))
I
(TT
Z)
+
Z:
Z4)ýh
_a1t,
d4-I
(yj-zj)'/j
3 4t
"(Y.
2 Y1Z
Z)
-I
(y4+Z#)312
ý
2.yz (y, z')- 7,
P
(
Y4+z
where
LZ3
41414
u, v, w are
the
physical
components
of
velocity
along
respectively.
Image removed due to third party copyright
Fig-(1-9)
Zamirls
co-ordinate
systemtequn.
(1.9).
30
Zamir
plane
of
considered
symmetry
y-z;
and
where
2
4-
w-0;
(1.12)
j.
and
his
the
(1.12)
+v
(181,
paper
with
U(x),
velocity
U(X)
A and
where
for
n>V3
Zamir
are
n
could
produced
showed
Equa
that
the
apex
of
a
fail
to
account
concluded
symmetry.
there
rectangular
that
(1.13b)
the
gradient
Implied
solution
by
of
free
a
n
Ax
(1.14)
computed.
He
found
In
a further
for
solution
were
equations
that
only
(191,
paper
the
very
solutions
case
1/3
nto
sensitive
the
used.
(1.13)
that
(1.13
by
constants.
be
produced
considered
a pressure
a numerical
tions
showed
disagreement
-
he
0
Ir
Zamir
given
technique
numerical
L
S
17
+a
as
Substituting
au
to ax
its
since
Infinite.
V(
=
symmetry
(1.11),
and
ap
as
of
was
(1.10)
zu
(1.13)
stream
Into
t9byde=0
second
equations
of
in
,4=0
plane
(1.11))
equation
ax
who
z/2
the
at
(in
uau
and
-
(1.11)
and
r2
ez
that
argued
coefficient
In
y/2
10)
put
YF2 ..
Ile
(ý.
equations
for
have
been
a
exists
sharp
such
Zamir's
criticiied
system
corner
1191
remains
as
has
to
correctness
the
near
these
equations
consequently
incorrect
to
1201
Tokuda
eddies
that
and
were
replied
the
of
Tokuda
eddies.
equations
Zamir
by
the
in
criticism
of
Zamir's
the
plane
and
method.
a)
31
The
matter
manner
however
may
it
and
is
for
necessary
(i)
be
the
corner
formation,
is
system
at
dropped
Zamir
for
dropping
This
term
(ii)
the
from
r
r
as
magnitude
derive
For
from
first
-
it
infinite
only
that3
ýVgl=
'
the
be
to
principle's
volume
derivation
0
and
the
used
r,
The
be
made
Zamir
continuity
the
at
involved
22
In
u1u
therefore
deleted
is
must
term
zero
element
0
(1.11).
term
makes
procedure.
should
0/0).
aW/8
to
compared
equation
obtaining
Zamir
incorrectly
in
in
equation
y-z.
an
this
justification
therefore
example,
prove
In
merely
was
invalid
an
to.
the
form
of
less
still
assumed
comparison
trying
for
of
the
course
term
this
(1.13b)
for
an
elemental
but
the
unfortunately
that$O%d4"assumption
purpose,
is
and
negligible
is
the
reason
(1.4)
11
trans-
terin
takes
at
very
although
condition
term
in
term
Zamir
This
In
the
e.
infinite
that
and
the
2/3
underlined
underlined
Assumption
deleted
the
unrealizable
zero,
of
A consequence
There
Indeterminate,
C
y=z.
this
for
term
ij
at
of
becomes
physically
be
a singular
(1.13a).
equation
the
is
(i.
y=z
J
Z
indeterminacy
(1.10)
in
( Y, + 7-1)
definition
the
rigour
is
at
coordinate
the
singular
Jacobian
(Y+ Z) G., -, Z3 )
=
=
by
are
2
9(V.&A. 43)
and
The
(1.9),
equations
a(ZOYOZ)
lacks
analysis
(1.11)
and
symmetry.
of
plane
followlng'direct
the
problem.
(1.10)
Equations
the
Zamir's
that
clear
in
resolved
In
22
uu
was
element
order
to
deriving
of
the
attempted
to
equation
plane
symmetry
implicit
the
its
defeated
32
component
parts
belonging
to
(iii)
is
the
of
the
same
Assumption
in
applied
same
is
manner
is
only
S2
constant
and
surface
S,
constant
coincide
later
Zamir
this
satisfied
It
is
the
chose
doing
the
use
(and
so
of
dering
is
admissible.
did.
The
the
brought
in
equation
consequence
defect
in
on
consi-
out
&;,3
the
have
may
a priori,
neglect,
are
Insofar
that
also
(1.4))
the
so).
assumption
Zamir
in
u
system
as
and
(1.4).
nearly
illustrates
this
momentum
of
(or
to
J=2,3)
general,
coordinate
isovels
the
the
assumption
the
the
when
incorrect
(i,
terms
assumption
a coordinate
condition
nevertheless
I
termsaUYag, V
of
for
used
true
in
untenable
surface
as
the
equation.
(1.5)
the
to
or
element
direction
is
which
c
3j
lp2l3t4)
=0Q=
01
Ia CL,
(311
33 12a
94A
63
?%,
521
P)
ul U3 + .13(3
35
incompressible
an
ýg
The
feature
of
viscous
term.
most
vanish
if
proper
derivation
which
complement
(1.4)
assumption
L. H. S.
of
suni to
zero.
the
the
of
the
continuity
53 ul
4-3jZ3912-
343
49ý22
(1.15)
5ý-v
and
the
orthogonal
defective
obviously
is
equation'(1.15)
Secondly
9"
943
1, +322
flow
steady
system.
coordinate
(8
Zp
2p
for
33a3
-Ff-T
+ -L
P
3
a9ls
of
absence
underlined
term
used
correctly.
was
equation
underlined
equation
(1.15)
any
would
A
introduces
terms
(1.8)
to
form
and
terms
the
therefore
33
flow
The
along
dimensional
problem
(i.
e.
in
which
in
adjacent
a plane
flow
problems
and
that
[2]
Zamir
experimental
The
corner.
like
shape
is
behaviour
leading
edge
Grossman
El-Gamal
the
flow
along
are
quite
the
difference
theoretical
show
Zamir.
symmetry
profiled
the
of
Fig.
(1.11)
from
different
leading
leading
in
plane
that
are
in
two
by
Rubin
distortions
in
shows
a typical
Zamir's
and
experimental
edge
edge
forms
used
whereas
the
each
El? -Gamal's
his
[201
El-Gamal's
Zamir.
and
isovels
in
of
results
the
with
agreement
particular
found
by
in
profile
El-Gamal's
in
Rubin
d by
development
velocity
results
This
Recently
and
Ghia)
the
of
results.
by
better
(and
isovels
reference
the
obtained
much
aerofoil.
obtaine!
corner
those
in
symmetry.
from
rectangular
sharp
streamlined
the
of
from
the
of
set
a zero-camber
bulge
in
from
sign
plane
The
the
a sharp
solution
no
of
was
experimentally
results
a problem
a right-angled
(1.10)Xtaken
determined
experimental
half
different
Fig.
along
model
the
across
different
is
treat
extensive
the
of
a pronounced
[111.
illustrate
to
applicable
contribution
an
flow
the
front
the
markedly
[4)
to
for
velocity
boundaries,
only
to
be,
must
plane
solid
used
produced
results
of
prevailing
flow.
also
showed
results
streamwise
to
[211
symmetry
the
be
vector)
conditions
vorticity
therefore
plane
velocity
analysis'As
corner
that
the
at
streamwise
cannot
the
of
in
happens
Zamir's
streamwise
the
of
flow
what
where
negligible
and
by
no
stream
independent
The
[4).
suction
The
is
affected
free
three-
essentially
be
can
the
planes.
e. g.
like
flow
an
poses
there
and
containing
the
is,
and
a corner
the
experiments6
are
case.
due
posbibly
Zamir
used
model'incorporated
a
34
Image removed due to third party copyright
Fig. (1.10)
isovels.
Streamwise
(Taken from ref.
t201
35
-J
24o
20.
Exncri-mental
results
Zam'ir
0
El-Gamal
0-:1
12.
1.0 0X 10
u
013
.3
Fig-(1-11)
symmetry
.4 .5G.
.7
Streamwise
velocity
plane.
09
10
in
2]
[4]
36
a knife
form.
edge
El-Gamal's
of
wall
His
.
suction
of
the
In
solutions
the
about
of
values
obtained
zero
suction
flow
of
In
regions
corresponding
The
perturbation
had
to
for
joining
the
As
equations
is
weakened
disregard
smaller
the
the
of
importance.
and
Carrier's
method
excellent
agreement
with
a rather
pleasing
approach
to
th'e
defect
of
the
re-valuation
corner
proved
is
problem.
the
of
large
solution
asymptotic
of
the
large
this
cross-
values
prompted
a device
as
be
to
extremely
the
increased
momentum
and
Carrier's
solut'ion,
is
equation#
vorticity
for
Indeed
III
and
for
method
vorticity
the
were
II
and
and
parameter
streamwise
parameter
small
This
streamwise
large
solutions
approximate
suction
motion
For
solution
domain
solutions.
the
between
coupling
i. e.
two
parameter
of
equation
dimen-
a
parameter.
for
overlap
proportion
the
of
regions
exact
of
by
solution.
suction
Carrier's
re-consider
satisfactory.
the
no
the
two
in
values
to
the
solution
parameter
small
no
independent
characterized
asymptotic
found
of
values
varied
Rubin's)
e.
parameter
all
him
(i.
El-Gamal
second
perturbing
suction
for
the
by
the
(. 1.3)
Fig.
was
was
For
parameter.
were
obtained.
of
flow
the
case
each
the
and
of
considered
first
The
corner.
although
He
work.
effect
rectangular
interest
some
present
coordinate
suction
sionless
of
the
studying
a sharp
distribution.
streamwise
,
along
is
the
to
preliminary
flow
the
to
suction
Rxi.
a
analysis
importance
kinds
was
on
theoretical
direct
to
work
of
values
produces
the
first
suction
in
results
solution.
boundary
of
This
layei
is
37
[22]
Weinberg
example
nearest
In
90
the
0
flow
as
course
sharp
in
long
discussing
as
the
The
taken
for
Apart
by
the
reference
1.6,
on
from
deals
considered
here
isovels
the
two
any
literature
problem
of
the
flow
along
the
work
now
to
(i.
the
e.
more
radiused
be
to
corner
can
4o
39
2e
le
0.
Pig.
(1.12)
4.5.
Pillet
Zo
geometry,
the
of
fillet,
velocity.
far
so
be
characterized
corners).
important
immediately
considered.
3o
(1.13)
Fig.
sharp
corner
so
local.
was
effect
It
5.
lo
the
literature
the
planes
a
the
which
on
that
streamwise
with-corners
of
for
of
work,
of
concluded
this,
show
apex
coordinates.
structure
confirms
to
its
across
compared
a
along
incompressible
stretched
flow
corner.
flow
for
the
the
on
Weinberg's
intersection
radiused
Weinberg
(221
only
a
placed
small
effect
the
along
the
perhaps
compressible
are
results,
absence
motive
t.
presented
from
is
calculations
remained
its
what
a fillet
and
his
be
to
work
the
af
having
fillet
flow
conducted
ý
dimension
layer
is
he
(1.12).
In
the
study
a corner
Fig.
to
his
of
corner
along
far
so
for
results
produced
is
a
primary
The
38
Image removed due to third party copyright
Fig. (1.13
Stieamwise-isovels
Xrom 'Weinberg
(.2ý1.
39
CHAPTER 2
THEORETICAL ANALYSIS
Introduction
2.1
The
most
in
system
coincides
with
the
desirable
that
such
away
from
this
provides
the
such
layer
whose
solution
Reynolds
The
found
that
In
are
frame
and
found
to
be
of
found
to
depends
corner).
with
the
on
boundary
e.
they
another
the
layer
do
into
Navierrespect
a boundary
type.
non-similar
reduces
not
type
non-similar
independent
for
conditions
(i.
expressed
local
are
the
the
a posteriori,
the
be
which
chapter
justifiable
of
into
developed
then
of
local
the
referred
parameter
is
defined
in-Chapter
then
derived
and
it
as
it
layer
corner
are
dependent
depend
far
form
convenience
layer
this
conditions
angle
the
boundary
boundary
the
of
of
the
is
It
because
which
into
corner.
plane
parameter"
the
surfaces
a cartesian
the
a curvilinear
coordinate
the
of
of
assumes
a form
but
framework
the
9corner
asymptotes
cornef
to
is
number
the
as
of
surface
approximation,
equations
one
equations
is
which
these
the
is
problem
a system
continuity
form
flow
corner
in
asymptotes.
a reference
A further
to
solid
treating
layer
and
which
symmetry
in
corner
Stokes
to
the
radiused
attempted
conveniently
coordinate
the
of
analysis
on
only
the
4.
is
on
curvature
40
2.2
Choice
The
for
of
coordinate
radiused
corner
following
the
which
system
a developable
constitutes
coordinate
is
system
surface
suitable
114
yXx3
22+X3
yX
f
3
yi
(i
and
is
systems
from
the
have
axis
in
x2=0,
A
an
orthogonal
a non-orthogonal
Both
angle
is
lt203)
x
die
A
Cos
the
is
to
y
2_
same
y3
to
is
the
the
curve
as
shown
in
Fig.
a function
of
x3
only.
plane
as
chosen
origin.
tangent
the
coordinate
X
-.
system
coordinate
curvilinear
the
y3
cartesian
system.
angle
measured
by
described
(2.1).
The
2
e3
Fig.
The
'(2.1).
Sketch
of
transformation
dy
coordinate
(2.1)
j
dx
j
is
(ivj
systeml
defined
=
112,3)
equn.
by
(2.2)
41
the
where
-ef
components
are
i=11f1
e,
.
The
12
10213
2
_tP
t2
3
11
o3
,, 203c0s
e
0
tensor
metric
2030
is
g ij
(2.3)
sin
defined
by
kt=3 kk
9
i
has
therefore
and
-ej
.E
ijk;:
the
"2 1
911
components
912
1
g22
Jacobian
The
the
of
913
0
g2 3
sin
g33
1
(2.4)
is
transformation
233
1.
-el. 3 -
i=
i.
j=
e.
2.3
Cos
Consider
a laminar
fluid
leading
the
If
components
the
v
a developable
and
direction
and
of
of
V
the
are
of
the
incompressible
an
surface
the
free
of
as
coordinate
stream
by
described
system
velocity
be
xi
be
in
the
at
x
the
respectively
velocity
Navier-Stokes
Ve j
0.
flow
steady
origin
edge
(2.5)
7io)
equations
along
Let
positive
then
(Cos
laXer
x2=0.
the
A
Boundary
viscous
eý-
-i
can
+
direction
in'the
vector
equations
contra-and
I's
jk
be
expressed
Itr'e;jk
co-variant
of
as
x
123].
(2.6)
42
where
before
a semi-colon
in
differentiation
V
is
the
kinematic
the
i
subscript
direction,
xi
is
p
The
viscosity,
denotes
covarient
the
and
pressure
continuity
is
equation
i
Written
(2.7)
in
fully
components
of
the
"a
IV
ýX-L
av,
vi
avs
aa: l
+V
3-V3
axi
+v2
three
(5ýýx
-v,
+vA
gvl
I
the
system
are
+ V, a
+v2
a+
coordinate
(2.6)
equations
IaV,
v+
xi
(2.8a)
(2.8b)
VAV
3 (a-r,
+ IV
8.v i -
Cr
x
Cos
Ik
-vl)=.
PaXJ
-2
122
COS'Au-
X
8X2
ý 'I's
Cr
3. X. 2
Gsn2, kL
Wý 2A
av
v
CT
2,
+
cr
+z
3wj
bi A
vs
C.
OýS;
L axi
C;
a
sea
X
COS 2.
A[2
.
jxjý
2
xi
(st
(75"
5w:.
cr)]
(2.8
where
2)2,1
[ a2
it
c=osx
C2
the
continuity
avL
equation
av2
+
(ax,
A (ax
Cr
OS
and
)
8V3
8,t 3
X, z
Cr sfnl
+
cog
a
X3
)2
(2.8d)
is
&
SInX V3=COS'A
0
(2.9)
c
43
Cr
Here
to
refers
the
curvature
wall
is
and
by
given
dA3
Cr
The
(2.10)
`ý
physical
components
191,
xi
of
of
the
A(i)
is
the
vector
the
g 11
this
=
v(3)
=
from
the
to
express
the
the
direction
components
physical
Vi
(2.2
'Jg2 2v3=v3
from
=1
there
is
no
distinction
the
physical
in
between
The
components.
of
i. e.
(2.4),
expression
covariant
are
quantities
obtained
relationship
(2.4)
To
9792
terms
V1
Using
on
in
components
contra-variant
in
components
sum
v2=v2
system
and
i
v1=
ýg
'ý g33
g22
coordinate
physical
(no
are
--"
=
IF9 11
v(2)
=
A
physical
vector
V(l)
since
9
ij
Accordingly
velocity
are
A
---"
-f9ii
where
tensor
a
of
=
and
(2.12)
V,
V(l)
v2
v-(2)
v3
923
derive
V
gjj
the
equations
they
+g
23
v(2)
boundary
(2.8)
are
6.
v(3)
(2.13)
+ v(3)
layer
and
equations
(2.9)
in
it
is
conveniýnt
a non-dimensional
form.
44
Uo-
Let
be
a typical
L be
velocity,
length
a typical
I
Re=
and
Considering
flow.
is
>>
LUoo/v
O(R
of
Reynold$
the
situations
new
of
number
lateral
the
where
following
the
e
be
variables
the
curvature
introduced,
are
14
OL
V=R, xýL
R XýL
= e
1/19
q'=R
ýZ3
14
z)/Uv
-v(.
=Re
ma)M
e
oo
.9
Uco
the
of
lowest
order
foll
Z.EV I
_
+;
Ts
ay
+
1
terms
layer
.
in
(2.8),
in
Rei
in
df
each
equation
retaining
gives,
i, I
(2
2
a
V,
9! E.z
,a
r2
COSA)
2-
v3
-ä -3
ael
g£A
q£j
Co
1
.2
5212A3ei
jx ý I
.
]
%.
jr
32
(2.15b)
Er
ir"
*e
21
.
15a)
az
a ZE-3
3 ý2
sinA
and
-2
'q
4-
3
L9'ýE
-2
(2.9)
s
equation
73 aq ri
ilt
-Re GL
'C.
(2.14)
boundary
owing
-
Jý
?+
+
4-Re?
I
Sub stitution
the
only
4ý2.1
-
12
3 =ýRe
;V
VU00
e1
V3
aý3
Co?, %
[2
ct
cos"A
2
+
Cr
+1
la£s
1
ccs
cog"z
'ý
ýr
)
a
( 6.;"
V3
'(2.15c)
and
ýgL
+9
9 jý-1.9
:rrZ+
-zz-
De)
593
C"nx
-
ür e3=
zý-o$
--IN
0
(2
.
15d*)
45
where
A
n.
-2s!
c a-V-3
+
CT
ý,
8
SinA
'FO-S-5
COST
dp
and
free
/dx
0
1
stream
from
found
a
(2.15a)
at
i2
is
assumed
distribution
velocity
consider
we
is
free
particular
(DIT
stream
the
where
0',
Here
known.
U(ýl)
velocity
defined
by
(2.16)
U(ý
m is
where
From
a constant.
U(R
Lim
that
requirement
(2.15a)
equation
1
)lUw
we
the
and
have
aei
-ei
e,
whence
The
development
are
most
(xix
)0
u00
(2.14)
variables
the
/
of
conveniently
the
found
were
helpful
clear
equations
in
expressed
in
but
terms
keeping
final
the
following
the
of
equations
variables
Is
e=
-n-L
(2 R*5
UE
q
where
-Ef
L
J[(,
oct3+
iIi
a-comma
differentiation
e. go
7n -i
E3,
before
with
Iz
eU
(2
_
cos
respect
5
(;
i
to
.
17a)
?n+ I
7n)u)
a subscript
and
e
(J=1,2,3)
refers
to
paýtial
46
01'r =
-P"-I
12- (E I)-
In-i
w
3
+ý 6jnj)ju
oc
((p
Sin
+V
(2.17b)
Vq*
2
-vn)(42ý5jnA +e)U_12(k
L(cpsin A+*)
=.
ir
The
flow
W(i)
by
0
variable
is
ILI
Sn" +U
dependent
function
Ii
of
in
variables
t21
streamwise
32
(A) (1)
All
the
to
related
(2.17)
S'M;L)U13
except
By
and
'-Z -7
X+4
vorticity
and
(2.17c)
X
are
definition
C171)
I
where
is
a length
In
geometry.
terms
(2.18a)
scale
the
of
the
characterizing
new
body
surface
variables
(2.18b)
where
RZ
da
Consequently
0
(-,n
&3
+
and
(2.19)
The physical
defined
by
(act)
where
W(J)
component
."i
43i,
tj
0,
1,
if
if
if
of
V
any
ijk
ijk
the
vorticity
vector
is
tj
of
Is
Is
Ij
an
an
*k
the
are
same
even
permutation
odd permutation
of
of
123
123*
47
On
(2.15)
substituting
we
get
(2.17)
expressions
into
equations
.
z
+
CP/, %
'-V
,,
13
+
Ujl
-c(-M+I)eu
"_
IZ +U
(2.21a)
4143
cN.,L6
3
+
(603 so, )+
.t'P3 +
VIC)
693
+
-
+
et4
(,'0 +X
e6
(2
tlo'L 2
+
(43+S
+
(S
(L+
3(-t
-K4
- IN-9) Url
-2
-m
(. L+414)tc
the
s
and
c
denotes
differential
sin
X
and
2
V
operator
cos
A
3c
i)
& Be
by
and
cross
differentiation
subtracting
terms
the
of
results
P'2
equations
to
give
-r-
and
1.
tL, j.
(2.21d)
vs a
P'3
(2.21b)
are
and
21.0
.
-
') 2
-" (a &ý)
a2
gradient
w))a
and
respectively
Fa,-il
pressure
(2
reads
I
The
uj, 3
3
-4
sa-,
-,
(zc
'P,,z
where
.
21b)
eliminated
(2.21c)
48
v0
0011 + I^OP3 +2uC(C---&3C,
43X
cse4
c
C'ý'j
3)E?+"(L--ul)eAecu
uc
+Se
-I.)2
C
_(E:
vif)
UU, l
32
4"6033
& C.
43& C#3)
"95
+
c+
+
is
which
(2.22a)
IU#33)
by
complemented
0
99
c
+
13
33
I
C13
le cu
+e
(2.22b)
c2
by
definition
(see
(2.21a),
Equations
the
boundary
lateral
The
layer
flowing
fluid
(2.21d)
for
a developable
curvature
O(R
of
are
evidently
incompressible
a viscous
by
characterized
finite
some
completely
govern
surface
at
e
(2.22)
and
development
over
equations
(2.17)).
equations
value
of
they
because
non-similar
a
contain
I
the
streamwise
stretched
9
implicitly
in
difficulty
associated
the
streamwise
value
of
with
direction
from
the
be
extremely
to
obtained
would
tational
terms.
the
equations
in
promises
-A
still
region
(2.18b)
solution
known
Fortunately
-=
to
a problem
the
that
O(R
at
a
This
in
realise
and
even
in
prohibitive
are
not
particular
solution
e
equations
an
integration
an
difficult
leave
is
requires
Cr
where
The
(2.19)).
and
non-similarity
a
and
explicitlyl
(equations
the
to
solution
exact
and
coordinate
if
compucompletely
49
By
intractable.
in
4
terms
reformulation
implicit
of
and
simplified'by
greatly
(which
the
are
The
verified
is
problem
components
explicit
trouble)
still
while
equations
in
this
simplification
of
consequences
the
components,
of
the
to
derivatives
explicit
the
source
character
components.
explicit
neglecting
primary
a non-similar
the
of
the
retaining
implicit
can
be
a posteriori.
i
The
to
is
method
terms
to
A where
containing
it
example
is
as
A refers
compared
negligible
to
u, v
For
or
w.
in
(2.21d)
that,
assumed
43kS(I-w))
Zcu,
A, l
regard
(I t ") c
uI>I
eupi
and
l(2.23)
2
)c
43
The
its
effects
definition
the
of
as
VP3
Jec
assumption
in
given
on
eu,
t
99 are
expressions
in
(2.22a)
deduced
j
from
(2.17):
e[ci-ln)ett,
L3
cpj
C), 3 99 *c
IU)
(-'&-z
c3
(2.24a
99
Similarly
e'ýh
3
-4
(I +
OC)l ;: Z;
(
-eKC)03
I
(L-, M)
(I -t ")
(2.24b)
143(613
+43r
__.
so
Evidently
the
in
assumption
in
non-similarity
itself
manifests
The
the
final
the
follows
is
k
of
layer
to
change
as
equations
presence
boundary
the
of
drastic
no
governing
through
form
what
the
entails
this
X13
and
be
to
equations
used
therefore
2
+ V" "03
Sotljpl
P.
42
CL - UZ)
[2
3 +
(C
eC,
(2.25
3) +
- -kn
eC(Ill
tý-M)
[2
WIG,
ut)
a)
'344kC3-US
)
X
'j,
3
)U13
ek
4'/C(10
+
99,2) +
3)
C
+
3
le c.
2 -m
V'*0.
5
Z
S9722
+
4-
-
-(
2.4. i
along
Corner
The
subject
(2.2*5b)
(2.25e)
ý0'
y93)
(1-#w)
le
a2
-4-4
Flow
P33
3
(1,
s)
tt
ki)4
e
-
3
2.4
CP)
0
(zc
'%tl 3-
+
(513
&3 Ae
+
tp". z
+
61
u-c2
(2.25d)
a corner
geometry
corner
to
is
defined
by
the
specifying
behaviour
of
satisfying.
symmetry
with
X
bconstant
a distance
respect
of
O(R71)
e
to
(say)
x
3
01
asymptotically
3
from x0
within
A
51
For
we
for
example
the
configuration
rig.
in
shown
(1.2a)
have
x30
at
as
x
3
-9
le
large
is
where
,
finite,
but
ý(2.26)
as
4
RE
x3
O(R
and
0
defines
the
corner
xC
2.4.2
is
to
in
of
zero
is
from
is
and
zero
The'
situation.
for
that
the
which
henceforth
attentfon
governing
(2.25)
equations
gradient
pressure
flow
corner
gradient
this
directly
obtained
and
case
pressure
confined
by
(2.27)
motion
of
important
streamwise
angle
7r
Equations
An
Ic
are
equations
on
m=0
putting
are
2
U+
19
41 Ull
+
002 + Yf Of
3+UP+
"1'
43
+
AC()6, ti, 3
+ 1?,2 -(34,5113
+
4,14,c2
(,
k'3)
e
4-4r
+
z)
.
,3
99UI) +
U. 3 +
(2.28b)
3
&-S) U0
(2.28c)
Vlo
3) +3
V+5
tt
+eS, 3, -244Lr-3)
+
I: X) =0
(Z C
+3-
3
3
& (UP3 3-U,
9ý,,z
-
51.3)
+2W?
="3
+
(2.28a)
rl), 3) 19
43)
99t2
0
tt'3
-
C20
=
0.
Cý+ '5 *)
&
(2.28d)
52
Utilizing
the
in
symmetry
the
problem
P.
be
to
are
in
solved
boundary
derived
conditions
Boundary
2.4.3
i)
no
in
the
subJect
following
(2.28)
to
the
'
section.
conditions
Conditions
The
<co
0
domain
the
3
equations
0
on the wall
condition
slip
that
requires
e=0;
U=v=w=0
(2.29
i, e.
and
from
a)
0
the
defining
(2.28d)
equation
(2.29b)
cz
O.
since
(ii)
from
=0
vector
3
are
on
physical
are
0)
the
continuity
the
symmetry
components
symmetric
whereas
with.
v(3)
and
plane
v(l)
and
to
respect
the
ay3
of
the
geometric
vorticity
velocity
symmetry
plane
&)(1)
of
y3
ci
the
variables
= v(3)'
= W(l)
involved
=0
in
(2.17)
(2.30)
these
are
conditions
=
v(2)
streamwise
aVC*z)
terms
0
i. e.
anti-symmetric,
In
(2.28c).
equation,
3
Conditions
The
(y
Z
0;
U9 3-ý,
9
u "Z
=01
S-.93 97 +6C, 'U
C
0
0
(2.31a)
53
for
A pre-condition
is
that
the
symmetry
coordinate
For
general.
symmetry
the
the
validity
equations
radiused
3
C=O
are
(2.31a)
equations
is
continuous
the
at
otherwise
defined
corner
(2.31a)
conditions
of
itself
system
but
plane
the
completely
(2.26)
in
the
become
03
U-43
0
(2.31b)
0
since
0
sin-A
The
case
of
a sharp
easily
dealt
system
shown
(infinite
corner
at
curvature
3
0)
is
coordinate
the
system
For
formed
boundary
the
by
more
surfaces
=
in
usual
(2.2a).
sharp
(see
the
vanishes
and
t4. -3
UIX
(2.2b))
Fig.
curvature
of
boundary
the
this
to
directly.
apply
corner
walls
the
example,
Referred
(2.31a)
plane
const.
for
using,
Fig.
conditions
intersecting
ý;
by
with
the
conditions
become
ý;
=0;
-S
=0
99"z +
4?
.
These
results
sometimes
with
about
the
the
plane
form
are
of
expressed
the
:=01
that
plane
and
to
is
of
that
provided
continuous
is,
it
that
they
flow
are
is
symmetric
Equations
symmetry.
coordinate
extent,
system
the
if
necessary,
not
coordinate
condition
3
y0
the
us
physical
the
represent
precise
they
to
=0
that
convenient,
respect
(2.31a)
remind
(2.31c)
is
symmetrical
satisfied.
system
unimportant.
in
the
which
54
I
donst.
const,
.2
Fig. (4.2b)'
Fig
sharp
(2 2.) Co-ordinate
corner
flow
systems
problems.
used: - to
treat.
55
in
Conditions
ciii)
I
As
free
the
c%O,
.4
stream
potential
the
vorticity
the
the
the
in
absence
only
the
must
of
a
physically
present
also
flow
condition.
independent
be
the
of
as
J;
becomes
limit.
this
is
this
gradient
acceptable
e
in
to
pressure
system
the
vector
velocity
streamwise
coordinate
from
obtained
respect
less
are
at
simply
with
e-i))
e---poa
V/
and
cross
O(R
(2.17c)).
equation
nonetheless
that
(to
90
on
are
requirement
In
vanishes,
(2.32)
but
invarient
'ý.
i.
C>O ;
conditions
obvious
its
approaches
velocity
(see
The
Ce-*-oo)
region
streamwise
and
value
ý;---->.
the
Consequently,
velocity
components
--, ý-W
3
The
variation
00
are
The
continuity
in
v
as
follows:
and
w with
respect
to
when
z
in
terms
of
found
-
equation
the
velocity
the
and
components
vorticity
definition
u, v, w are
3
t1#3
C+W+s
d),
where
In
the
uu'2
whence
(cw),
IT)
-j-j
=-2
limit
`2
't'
-I'
PZ
f,
-
co( i)
'4"
(IV
-
+5
(C W)-#3
'El
c
-W)p 3
J
-
as
'3
(2.33)
Sl
W' 2 ý'
=
56
and
(v+sw)
=
,, 3
i,
cw = const.
v+sw
From
the
(2.34)
= consto
(2.30)
condition
symmetry
have
we
z3
CO.P
and
1;= 0; -W=0
0
then
c 34
since
00
(for
w0
(2.35a)
all
Therefore
v=
is,
at
const.
by
boundary
layer
the
as
and
say,
unknown
to
need
3
(iv)
(2.35b)
from
outflow
be
will
value
whose
the
with
for
the
below).
Cp
and
I?
follow
now
as
99
C>0
constant
(Art.
conditions
(2.35)
(for-all
match
o oo
boundary
(2.17)
from
an
present,
furnished
The
A
30
ý4>,,,
C
13)
(2.36)
3
Y/
Conditions
These
are
(2.28)
equations
3
the
large
by
the
following
as
found
from the
3
1;
as
--s-oo
dependent
flow
It
is
of
analysis
asymptotic
that,
assumed
variables
can
be
for
represented
expansions
..........
Equations
(2.37a)
[101,
Rubin
see
are
similar
following
the
to
those
'
page.
used
by
Pal
and
57
OCP
Uwe)
99 -K
n=o
tw
0Z
which
are
v~
ýo
10
Vve)
00
E
W, ý
to
AOL
labour
all
is
in
those
is
the
f or
large
equations
(2.37)
With
3
ý
this
(and
X
*)
)u,
+ýs
algebraic
the
problemp
unaffected
of
by
motion
which
this
conve(2.28)
-(
+V0
(2.38a)
9,1) + ZUefetl
3
4PP2
g"ý)
+,
the
become
e(CP 0-2
+ lý
( ý*cp"
of
equations
CP LL,
2
but
choosing
quite
the
choice
expansions
require
physics
is
VU+
*2
70
on
essential
interest,
primary
nience.
of
eased
The
A
functions
considerably
expýonentialiy.
3
(2.37b)
nO0
similar
e-rt+
(2.37a)
19
by
complemented
general
n=cý
00
C9
o',(e)
00
F,
In
(pace)
-n+I
n=O
-n
j
_ct4J.
V-*3
-
e'-,
?'0U2:
(2.38b)
(2.38c)
--
(2
-e2a
where
43
and
sc
Placing
retaining
equations
respectively
are
series
terms
of
(2.37a)
equal
sink
into
order
COSA
equations
3
ý
in
gives
(2.38)
the
and
following
.
38d)
58
order
zeroth
uöl
gl, u01
+c
(2.39
.=0
'
e',+C-tr c
4ýAP,
9, +Z0
(2.39b)
+Z0
(2.39e)
e
Order
nth
Un
(n=1,2,3,...
+CU.,
00
An
Eln +CU
[C
(1-TO 1ý,
4- +2
4-2C*
(2.39d)
)
ýh -22C,
+C
Z
Uo
Up") U., ++C
U,, Un' +
zCU;
(2.40b)
0
l
.,*2 an
a prime
denotes
(2.40a)
80
0",
where
a)
(2.40c)
Cis
differentiation
with
respect
1) un"
+
42.40d)
to
Here,
A.,
n-
1) u.., t
4-
U
(n -Z) On-i
n-
+
[(n
Uv
Uo
C*S*
I+
+2
t9,
l. i
ui
*(n
(2.41a)
Vui-l
un-, +1
+2 uj U'n-j +
un l-j+'zC-
& rol
U;
-i-1) -jU&j-j
(2.41b)
and
71-2) Yn., 1
rPn-1 +
(11-2)
cpn-,z
(2
.
41c)
59
Sums
with
defined
as
In
than
in
one
(2.41)
equations
are
zero.
to
addition
following
the
less
limit
upper
(2.39)
equations
complementary
zeroth
(2.40)
and
equations:
one
obtains
-
order
a
=
(2.42
IuG
=
(2.42b)
order
nth
z
*(
V
un
by
(2.37)
substituting
The
(2.39),
series
(2.37)
boundary
(2.35)
and
(2.40)
into
(2.36)
(2.42c)
-wa)
-
(2.42
into*(2.17).
appropriate
equations
t2=
a)
(2.42)
and
boundary
to
of
conditions
are
the
system
by
found
placing
(2.32)9
(2.29),
conditions
of
get
0
Wl
0n0,102
904
22
--)woo
W't
o
0C
Wn
The
uo=f
substitutions
identicallyp.
OCP=c
(2.39a)
transformA
to
f
///
2:
r: E/I
(2.43)
19,0
=
(n=1,2,3,...
=0
f,
and
which
its
satisfy
boundary
)
(2.39c)
conditions
d)
60
.Z
0;
f
00
f
and
is
This
known
details
full
plate
here
omitted
(2.39)
equations
in
problem
Its
system.
are
of
solution
(2.44)
flat
coordinate
the
and
0
Blasius
the
cartesian
oblique
The
just
fl
is
solution
(see
an
well
(241,
ref.
223).
p.
therefore
are
Ucf
0
(2.45)
f
and
from
cf
(2.42)
V.
f
(2.46)
-f.
0
Consequently
(v,
Lim
)
f
Lim
(2.47)
f).
0
First
order
The
(n
equations
1)
for
equations
u
and
are
It
#
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C
2-
with
--j-
The
only
boundary
(2.40d)
00
1P,
0
00
possible
conditions
A, =O)
(si-nce
is
this
of
solution
rVj
U,
=0
system
of
Conseque
equations
.ntly
and
equation
yields
(2.48)
61
on
which
in
substituting
the
equation
governing
0
for
+f
(2.49)
C
The
boundary
conditions
gives
are
0
00
(2.49)
Equation
has
(B))
Appendix
this
(see
(2.48)
ea
in
13
C
V,
Efl
(2.50)
yields
z
f 1#1)
(n
equations
order
4%
-it
+SC
18
result
61,10
Second
solution
all
-/3
ff
C9
Placing
form
closed
2
3
W0
the
=
+
(2.51)
C*
2)
Here,
ABC
222
It
can
-L
C*l
from
shown
(2.40)
equations
and
(2.42d)
that
algebra
some
with
be
=0
W2
0
W,
+
with
20W
(2.52)
=0
W
00
Like'
for
it
the
is
for
equation
which
found
the
,2
only
that-u
rr)
y;L
possible
22
(P
;7- 0
this
is
a homogeneous
solution
0'
is
w2m 0.
In
general
problem
Similarly
for
n >,,
2
62
-const,
(Presel
coo- -di
systen.
nate
Cý
AlGeb=a1c
dec- "Y
=C,., Ion
linlt
AsynDt'Otic
:ro=
Fig.
of
Rubin's
decay
exponential
..
c Oflsý,,
(2-3)
systems
co-ordinate
algebraic
illustrating
A sketch
decay
of
the
the
with
respect
flow
variables.
role
to
coo=d:
Vnatc--
63
(2.40)
equations
and
consequently
The
factory
This
re sult
since
and
a
role
is
The
there
Since
region.
bringing
valid
Knowing
that
to
map
would
3
ý
the
the
infinite
without
the
no
provides
the
the
Rubin's
the
condition
in
whereas
solved
in
of
on
the
line
any*ý
potential
mechanism
other
is
phenomenon
series
appear
first
is
decay
algebraic
range
In
therefore
are
--4-oo
(2.39)-(2.42)
considerations
infinite
results
3
E;
at
strictly
equations
at
the
and
be
decay,
algebraic
Rubin's
3
condition
series
in
and
is
as
such
evident.
line
to
their
and
(2.37)
decay
equations
assumed
the
out
entirely
missed
use
the
form
a
expansion
algebraic
used
systems
algebraically
of
is
been
have
we
result
symmetry
boundary
a
of
the
the
the
use
the
system
on
acknowledge
the
be
present
is
and
The
1111
Grossman
coordinate
(2.3).
Pal
by
analysis
paradoxical
Fig.
could
satisfactorily
I
decay"that
different
satis-
conditions
the
Rubin
the
eminently
coordinates;
by
algebraic
for
the
an
that
necessarily
coordinates
permit
only
must
boundary
must
present
fact
apparently
in
and
one
system
the
consequence
coordinates
for
this
region
shown.
of
.'
crossflow
into
boundary
the
solutions
the
be
that
of
the
decay
normally
values
an
illustrated
decaying
ýý '%ý
in
exponential
means
for
exhibit
in
part
an
it
Evidently
play
terms
the
and
[141
seeking.
algebraic
would
not
[101
Ghia
and
it
were
Rubin
the
moderate
at
is,
solutions
Uft
imply
to
layer.
applied
that
of
seems
boundary
= In
the
v=wn=0.
vanishing
components
be
have
On
finite
at
any
to
preclude
their
into
we
and
may
not
practical
application-
a transformation
introducing
of
there,
Sý
the
finite
range
of
a
64
new
to
An
variable.
seek
a series
behaviour
is
to
and
to
the
numerical
conditions
are
for
them
profitable
in
they
this
situation
are
---*P-00 ;
/I
3f
11
a
I-A
cSi
+
-*
+s
c+
C%3
The
equations
(2.36)
(2.32),
(2.31),
The
the
it
of
method
studying
is
layer
boundary
by
specified
(2.53)
C*
cp
but
alternative
The
applicable
convenience,
the
later.
given
directly
is
and
scheme
results
flow
the
represents,
immediately
most
mapping
the
collecting
t
The
is
difficulty
the
through
better
that
3
employ
obtain
asymptotic
way
expansion
large
at
approach
used
alternative
nature
flow
along
(2.28)
and
solution
boundary
adopted
decay
the
as
emphasised
a
is
a corner
completely
(2.29),
conditions
(2.53).
and
of
S*c
into
matter
the
avoids
of
for
boundary
layer,,
importance
that
the
some
need
3
the
decay
This
is
of
the
is
crossflow
demonstrated
by
the
in
algebraic
analysis
of
the
both
and
following
section.
2.5
As): mptotic
To
show
decay
the
the
of
crossfiow
decaying
algebraically
of'the
character
2
crossflow
large
it
is
to
sufficient
arbitrary).
This
consider
simplifies
the
flow
the'analysis.
for
65
If
it
is
the
Consistent
is
decay
is
the
corner
for
algebraic
settled.
series
that
shown
it
then
is
Is
as
this
with
in
algebraic
a
this
region
the
and
whole
the
observation
matter
following
chosen
j.
+I
-I-
^
+-,,,
(2.54)
A3
vn
O, lt2,..
321
and Wn (n
)
functions
are
t+
Op+ sinA
only
of
for
large
and
for
A >0
( 7c >
and
arbitrary
Ac, > 0)
'S
where
(2.54)
Equations
are
continuity
equation
(equations
(2.33)).
The
to
be
and
is
series
<0
for
sin
designed
(17t
in
used
conjunction
the
definition
the
satisfy
Jt)
with
e vorticity
streamwis
to
> xc>,
asymptotic
conditions:
f ixed,
ý___,,
v
- 0,0
vo
'A
equations
WW0
fixed
but
large
v
ww0
The
analytical
simplification
from
stems
large
whereupon
equations
(2.33)
reduce
the
(2.35))
-10.0
^v
0
A
the
U-1
to
(see
fact
XjO
approximation
that
for
66
(cw)
v 02, +c
0
, (2.55)
(w+sv)
Substituting
terms
2-
(2.54)
series
the
of
(Bw+v)
same
in
order
t,
30
into
the
(2.55)
following
and
re. taining
equations
are
obtained
Zeroth
order
A
(CW
0
s^
First
(2.56
11
w +V
order
cw
(sw
13
(2.56b)
A
+V
=-O
3
1
1),
.0
Second
order
1A
-V
1c
+-
cw
(2.56c)
-tsv
A3
+w
)+t-
S
(V
The
boundary
30
2+sw2
conditions
3
ý=0;
are
A
wo
Integrating
=w2=0
A
vo
at
Equations
= W,
(2.560)
3
and
+SW
v
(2.56a)
have
equations
the
solution
(2.56b)
A
cw,
v
0
13 ,w=o.
gives
constant
(2.57)
sw+V,
=
constant
67
Like
(2.34),
equations
the
A
V,
From
is
constant
(2.56c)
equations
(2.57)
of
solution
wl
we have
integration
by
3
A3
Ic
w 2c
3
49
c-c)dt
+0
(2.58a)
consequently
and
C*)
+ sm
c
fo
C _C
ce
The
6
constant
by
using
an
to
find
[11]
Terms
if
the
show
the
and
to
the
that.
in
employed
encountered
solution
numerical
on
reference
the
solving
right-
problem.
higher
The
in
order
terms
(2.54)
series
however
obtained
nature
algebraic
3ý
decay
the
of
are
to
sufficient
the
of
determined
be
can
in
crossflow
direction.
(2.54)
Series
not
constant
corner
of
desired.
is
the
from
similar
approach
sharp
angled
determined
be
can
t3
the
being
while
form
only
that
adequate
may
be
used.
for
its
For
example
purpose
the
series
A
JV
titX+
and
(2.57)
and
for
analysis
terms
solution.
arise
(2.58).
Pal
w will
in'the
use
fifth
of
also
Rubin
and
corner
and
equation
the
yield
1101
higher
(2.59)
i
solution
showed
problem
(2,59)
VA
.+
-ti
9 D0 sharp
the
The
++
for
form
a similar
Untý
that
order
terms
or
indeed
in
an
asymptotic
logarithmic
in
a series
a linear
68
combination
the
of
appearance
detail
is
2.6
the
The
on the
is
result
be
v
w at
evident
oo
together
a
matter
whichever
velocity
w
'is
found
from
the
definition
with
the
shown
by
referring
case
system
and
normal
an
the
wall
the
coordinate
of
the
corner
surfaces.
components
of
the
wall
represents
to
and'Z
the
This
orthogonal
the
perpendicular
coordinate
stretched
to
parallel
to
references
corner.
sharp
(2.60)
of
one
respectively
are
rectangular
where
in
derived
that
equation
constitutes
w
a
(2.60)
ýb
-
of
of
f)
.
generalization
in
and
Aef
lf
--(-oll
coordinate
vector
3
ý-
di
a
large
(at
is
crossfiow
(2.53)
and
[131
cartesian
If
d)
/31
cs
can
is
This
be
to
and
asym2totic
velocity
(2.42b,
[91
decay
anticipate
suitably
behaviour.
algebraic
crossflow
equations
This
might
used.
Remarks
It,
(2.59)
and
logarithmic
of
only;
series
of
(2.54)
of
the
then
wall
crossflow
the
we
have
Cos
vv
Cos
(2.61)
Substituting
v
ww+v
into
(2.46)
sin
and
(2.60)
gives
77
(2.62a)
J30 t an A
w
where
F(. n)
a dot
system,
and
is
is
given
the
denotes
by
Blasius
function
differentiation
inýthe
new
with
respect
coordinate
to
69
13,
=Lim(
11-f»00
kn
7Z
(2.62b)
(*Z)
1-F
ý
in
or,
terms
the
of
system
coordinate
by
variablest
It
(2.62c)
cosA
Equations
flow
(2.62)
velocity
depends
and
45
angle
corner
the
is
vector
variations
same
clearly
that
independent
of
show
only
0
and
when
velocity
crossflow
the
on
the
the
asymptotic
wall
corner
For
0
yields
[9)
references
90
a
(2.62)
in
in
obtained
curvature
angle.
substituted
cross-
and
(131.
tacit
development
the
Throughout
w* /V
that
assumption
far
so
0(l)
(for
been
has
there
0
7Z>
the
It
is
N
in
the
condition
to
points
will
require
The
the
past
(2.4).
w
example
the
distribution
image
in
the
of
w
that
with
boundary
layers.
(2.622)
can
which
problem
as
such
be
that
given
flat
infinite
quarter
and
3600).
or
(12].
Stewartson
as
written
(2.62d)
of
and
changes
is
sign
for
w
+450.
influence
the
00
HC*J)
of
to
a
[251
Howarth
therefore
for
respect
controlling
of
independent
Fig.
in
in
formulation
A, tanX
are
XC~
character
from
w
for
violated
angle
side-edge
and
velocity
HCJ)
(corner
alternative
w
and
0
be
will
changed
Stewartson
by
plate
a
some
flow
the
90
of
vicinity
This
to
this
that
evident
explained
of
the
is
potential
a juirror
antisymmetry
remarkable
when
shown
For
with
-450
This
is
H(J)
we
flow
understand
on
the
70
oý%
v
m
0
0
0LL
ci
(c
Co
'i
71
/,
A
10
X2
ofoutf
outflow
u
magnitude
ag
Xx
ýOmRe-lx
Fig. (2-5a)
4
Fig. (2 5b)
72
The
corner
in
terms
which
domain
the
to
wall
at
each
to
O(R-i)
ex
in
potential
the
0+,
the
from
outflow
parallel
a vector
is
This
the
layer.
In
and-450.
'walls'
the
is
these,
indicated
jx3j'=
20+,
of-x
vicinity
to
this
is
corner
at
solution
the
crossflow
(/3, /Cosý)R.,
+450
flow
normal
of
= 0+
the
the
potential
R -1
ex
jx3j
is
potential
resultant
cases
When
between
magnitude
crossflow
(2.5).
Fig.
the
matching
The
controlling
now
coordinates
and
The
x2=
of
region
component
0+.
0-4t;.
physical
vicinity.
region
crossflow
and
flow
the
jx3j>
plane
Consider
in
has
the
symmetry
potential
the
this
x2=0+,
wall
the
0+
corner
in
effected
solution
unstretched
the
of
the
over
jx3j
x20+,
be
to
extends
the
of
region
viscous
is
layer
I Je1<00
is.
0+
z
into
transformed
is
result
the
but
(2.6b)
This
method
Method
in
transform
which
by
relaxation
physical
component
the
w
to
in
wall
direction
the
in
is
the
of
the
of
(2.6a)
figures
behaviour
antisymmetric
of
have
been
for
conditions
directly
oo
to
used
on
the
knowing
at
arrive
corner
layer
the
at
H
solution
oi
(2.28c9d)
their
these
are
The
parallel
could
Equations
solution
(2.6).
opposite
<c,. o
oo 4ýr
above.
boundary
asymptotic
z3
t;
ý
--- 3, c%O and
2.7
Fig.
illustrates
and
mentioned
0
region
crossflow
magnitude
and
w
in
shown
potential
same
the
present
equations
generally
methods.
are
form.
into
better
It
is
elliptic
suited
This
for
unsuitable
is
for
effected
more
numerical
Poisson-like
numerical
by
to
convenient
formi
treatment
differentiating
73
both
equations
work
so
to
respect
with
L 13
ý
-(
C2e,
-4-
Jy
3+2CC, 3e -(6C'31C
63(
t3+
(C
$13
C G,,X; -
I- I's,33
,ýPgz
6
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22
1193
The
(2.63a,
numerical
b)
stretching
infinite
domain
larly
directed
This
component
in
(2.28a,
equations
in
effected
the
2.63b)
b)
and
quarter-infinite
known
to
It
e
--j-po
can
problems)
be
imposing
4
moderate
value
of
streamwise
known
solutions
their
vorticity
the
boundary
done
ý
as
velocity
for
,
the
has
special
been
in
boundary
at
conditions
say)
little
with
by
verified
case
of
the
the
boundary
the
CWax
(=16
values
into
that
commonly
u.
component
slowly
more
is
particu-
potential
therefore
(as
replaced
by
to
likely
quarter
is
velocity
considerably
is
the
map
streamwise
asymptote
but
layers.
the
require
would
Interest
streamwise
with
done
to
domain.
the
together
exponentially-fast
boundary
both
at
be
to
this
23
ý
and
-a finite
into
here
are
component
with
For
00
variable
the
0
3
D<
region
be
strictly
should
(ZC+
3 3)
of
solution
.
63a)
yf"g 3-
eý-.
3 C13 +
U,
3) Z
(2
3
26
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50
Y;
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-
on
algebraic
some
plus
90123- 603(PP2
qcl)
layer
and
that
CP033
at
4
a
effect
comparison
0
90
sharp
74
To encompass
corner.
it
layer
the
boundary
the
transformation
the
decay
was
deemed
in
flow
the
into
variables
incorporate
to
necessary
3
(2.64)
a+ b
maps
where
a and b are
maps
equal
in
intervals
3
ý.
in
growing
size
effect
increasing
ý3
to
respect
A
derivatives
[26,14,16).
bounded
thatIV
0
and
is
difficulty
like
(2.64)
It
the
are
is
easily
is
by
desirable
the
changes
In
rapid.
applicable
3
ý
so
transformed
from
this
the
only
if
that
their
(2.58)
equations
3
ý
to
the
become
coordinate
respect
with
removed
has
0 where
large
evident
unbounded
It
3
a
for
bounded
transformation
3
in
with
intervals
be most
to
is,
respect-to
with
This
near,
resolution
<-11b
into
chosen.
increases.
in-3
are
variables
4.00
unequal
expected
transformation
dependent
t2
as
are
interval
the
vicinity
into
the
of
be
to
constants
step
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endix
Chapter
(C)
4.
78
CHAPTER 3
EXPERIMENTAL
WORK
available
for
Introduction
3.1
the
The
experimental
results
of
Chapter
2
is
these
two
what
appear
our
be
to
little
interest
corner
and
experimental
data
experimental
phase
directed
towards
Surprisingly,
a lengthy
a range
of
different
view
in
this
report
to
the
out
its
the
corners
covering
time
geometry;
experimental
over
has,
along
on
is
there
no
The
been
therefore,
model
corner
of
a period
the
a variety
indeed
a
information.
precluded
work
practical
flow
a suitable
stretching
theory.
situation.
missing
of
the-
supply
corner
the
work
this
manufacture
overwhelming
this
present
in
the
sharp
4
their
extending
is
for.
enquiry
knowledge
for
available
root
the
obtained
sufficient
therefore
writer's
expense
corner
of
for
business
This
carry
data
For
have
present
or
works
existing
of
the
repeating
providing
the
months.
to
in
[3)
data,
of
in
corners.
sharp
Barclay
and
context
reference
radiused
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These
central
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merit
experimental
0
135
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presented
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measurements
Of
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comparison
analysis
0
90
EI-Gamal
cases
was
theoretical
the
limited
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work
construction
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angles
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a single
importance
several
obligatory
model.
the
In
corner
El-Gamal's
tunnel
in the
at
obtained
wind
were
results
University
London,
location*for
the
College
the
present
0
that
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Elhave
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test
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same model
work
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79
ang le
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Limiting
is
corner
less
Together
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of
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corner
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3.2
)
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on
view
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experimental
[1,2,3,4p
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at
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defined
x3).
is
root
(2.1)
equations
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Consistent
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A=
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attention
conditions
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his
transverse
to
subject
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way
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appear.
model
Corner
The
to
will
This
contributor
known
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Corner
3.2.1
every
in
was
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provide
theory.
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programme
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reflection
271)
of
radiused
single
at
we
to
corners
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data
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sharp
test
adequate
it
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markedly
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to
work
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restrictive
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90
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about
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possibilities
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x3=0
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continuous
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3
),
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equation
properties
everywhere.
80
The
offered
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function
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3
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limits
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at
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to
large.
very
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two-dimensional
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above
boundary
82
layer
13.
thicknesses
Omm.
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of.
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In
academic
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vicinity
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approximately
programme.
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the
were
means
extremes
experimental
choice
large
small
or
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be
to
that
values
greater
of
practical
importance.
In
discussion
r0
consistent
large
been
it
be
3.2.2
strong
Model
design
corner
extensive
the
These
corner
to
extends
layer
to
adjacent
boundary
layer
or
on
four
the
the
the
edges
plates
theory,
extent
character.
therefore,
to
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give
For
square).
of
corner
dimensions
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with
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boundary
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plane
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although
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width
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symmetry
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x
accepted
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flow
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only
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Omm was
experimenter
long
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scope
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least
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The
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10.
Preliminary
one.
the
that
accurate
hoped
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Workshop
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was
anticipated
might
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since
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corner
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11.966
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14.872
16.304
17.729
19.150
20.568
21.984
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0.178
0.723
1.562
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3.881
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6.408
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9.156
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16.185
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Table
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Coordinates
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84
region
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was
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It
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sheet
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best
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sections
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Fig.
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x
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The
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CORNER MODEL
89
The
method
unknown,
writer
the
Departmental
and
are
we
instead
benefits
A
is
of
than
still
to
corner
without
subdivision-of
To
sections.
of
the
used
to
generate
the
mill
technical
staff
methods
the
use
of
without
incurring
has
form
the
of
construction
C1
OfAspade-cutter
would
end-cutter
this
of
the
reap
their
shown
in
the
corner
the
of
for
the
need
the
corner
form
into
it
a corner
for
Spade-cutter
the
if
first
a
the
view
properties
spade-cutter
and
In
which,
cutter
end
the
advantageous
life
profile
types
a cost
expensive.
length
any
on
be
an
has
able
profile
would
an
at
cheaper
somewhat
cutter
being
two
relatively
manufacture
spade
the
A spade-cutter
It
though
with
that
-cutter
or
last
the
of
great.
alternative
opinion
cylindrical[,
disadvantages.
to
discussed
the
of
but
very
in
resulted
outlined
corner,
been
Workshops
a
sketch.
have
again
now
of
model
must
has
the
just
manufacture
satisfactory
a completely
though
of
an
ordinary
approximation
spade-cutter
used
was
end-cutter
only
to
the
to
form
first
required
finished
the
surface.
Experimental
3.3
It
arrangement
has been
corner
was to be set
[4]
in
study
corner.
available
his
The
for
of
robust
the
that
remarked
the
the
measurements
the
flow
radiused
in
wind-tunnel
had already
support
new
and
a long
system
corner
which
been used by El-Gamal
a sharp
for
that
and
rectangular
corner
was
was
therefore
used
90
for
to
this
its
purpose.
and
support
the
essentially
is,
of
in
fact,
The
situation
of
its
position
within
little
very
in
the
El-Gamal.
rational
There
positioning
as
[21,
Zamir
by
were
wind-tunnel
by
a wind-tunnel,
used
arrangements
similar
in
choice
corner
a streamwise
the
respect
with
corner
described
that
as
same
the
the
witness
[31
Barclay
very
and
El-Gamal.
(3.3)
Fig.
of
plane
edge
0.27m
downstream
the
section;
of
plane
the
0.2m
The
be
varied
clamp
drive
on
system
drag
the
turntable
the
corner
the
was
encased
corner
the
minimum
stream.
the
and
the
chosen
thin
tunnel
unwanted
the
was
on
floor
which
yaw
angle
for
the
corner,
of
plywood
the
handwheel)
floor
formed
so
that
a rigid
disturbance
the
corner
to
worm-wheel
support
forming
the
the
corner.
a very
the
sheet
a
the
of
and
the
between
pitch
carried
Finally,
gap
corner
the
removal
immovable.
in
the
mechanism
(and
wind-tunnel
of
of
was
corner
tunnel.
wind-tunnel
pitch
spanning
aerofoil
within
on
the
rendered
itself
the
been
the
test
upstream.
the
of
permitted
adjustment
had
a position
system
with
O. llm
edge
of
support
flush
fine
very
effective
the
and
set
allowed
Once
support
lower
floor
the
above
corner
turntable
The
leading
the
Of
plane
therefore
was
the
the
and
inlet
the
geometric
with
coincident
wind-tunnel
of
edge
plane.
-exit
nominally
to
trailing
the
of
The
position.
is
corner
symmetry
was
in
corner
the
of
symmetry
vertical
the
shows
a low
underside
of
entire
structýre
model
incident
offering
air
91
cii
Itl
c31
-
Z
*Z
11m
.
l.
cia
-
93
_a
po
cu
10
-
2
-
.
Iti
0
71
92
The
flow
hot-wire
temperature
an
rms
voltmeter
and
scope
was
an
help
to
used
low
anemometer
bridge
balance
turbulence
in
flow.
The
wire
having
a5
PM
shown
in
the
probe
(Disa
type
The
55A25)
holder
probe
The
is
wall
[31
in
wall
the
their
tite
re-aligned
on
expectation
to
its
the
as
rotation
re-alignment
of
with
involved
What
in
general
dimensional
if
the
axis
flow
the
the
be
only
regions
plane
and
parallel
of
the
the
symmetry
with
the
In
the
of
the
plafie
other
as
The
is
trouble.
if
the
evidently
the
notbe
should
wire.
implies
problem
in
words
well
is
wire
symmetry
walls
Further-
again.
tolerated
this
the
precisely
system
of
be
contrary
return
at
might
at
and
fan
symmetry
flow.
down
traverse
be
to
It
plane.
holder.
horizontal
has
the
that
to
wire
probe
is
corner
the
always
but
the
three-dimensional
point
it,
the
to
on
the
the
of
after
isovel
are
wire
Barclay
the
shut
not
centre
required
really
isovels
are
fan
of
symmetry
are
we
to
the
wire
the
of
tangent
adjustment
the
at
symmetry
corner
promises
of
the
the
and
parallel
since
does
the
[2)
orientation
switching
of
require
eccentric
physics
on
value
wire
system)
speed
length.
to
respect
with
nature
wind-tunnel
air
miniýture
1.2mm
and
Zamir
through
the
previous
also
manual
the
of
Fig.
This
tedious
passing
the
diameter
of
presence
was
probe
oscillo-
the
of
the
of
warn
the
probe.
a
that
to
the
already
(with
requires
had
An
adjustment
consideration.
experiments
traversing
that
for
to
aggravate
so,
of
a matter
will
as
orientation
nearest
more,
is
to
and
voltmeter,
a square
filters.
pass
a fine
attain
D-C
incorporating
unit
and
constant
a digital
with
auxiliary
high
and
a Disa
was
anemometer
generator,
signal
instrument
measuring
plane
the
and
twoa horizontal
93
-To traversine
cear
To ane=OMOUý=
bar
SuppoxýUng
5a. Proba'cable
To electr1c.
inclicato--
-l- contact
Se +1 screws
ý7Suppox'-U'ng
ba==*ol
ba: r
Supportina
Pxobe
cable
Insulated
Wire -
I
Section(A-A)
Short
probe
/000'
cable
Locating
Pmobo
ilnsulai-
cil thin
Probe
Wire
Scroll
nupport
boay
Sensor
.
r,upýoxt
Electrical.
contac: t pin
Fig.
(3.4)
Probe
holder
construction.
94
is
wire
the
optimum
a horizontal
wire
the
two-dimensional
the
wall)
troublesome
out
his
the
The
that
for
the
wire
was
completely
findings
in
and
the
is
tion
re-alignment
by
Fig.
point
of
from
distance
irrespective
wire.
steel
The
pin
needle
commonly
on
point
used
to
the
fasten
by
orientaof
a
seem
to
involve
in
each
end
being
sheets
on
Contact-was
The
arrangement
that
ordinary
of
the
vertical
with
wall
an
made
a point
and
the-same
of
traverse.
respect
with
wire
contact
electrical
the
El-Gamal
case
advantage
establishing
was
wire
lamp.
at
of-
by
adopted
axis.
the
always
basis
the
probe
a small
made
angle
horizontal
contact
the
at
conclusively
once
the
of
probe
has
is
wire
wall
the
It
-
just
electric
the
configurations
present
to
of
(3.5).
the
an
attached
ignition
the
of
on
wire
would
point
below
the
in
shown
centre
point
vertically
registered
is
centre
and
a single
the
than
more
based
was
a needle
plate
in
of
to
parallel
on
was
alternative
wire
the
of
wall
between
the
the
of
the
to
the
since
Location
use
welcome
corner
is
wire
The
work.
the
the
It
a horizontal
present
time
showed
interest
of
to
boundary
two
the
satisfactory,
that
the
El-Gamal
wire
and
numbers
particularly
radiused
for
results
Reynolds
every
the
indistinguishable
virtually
for
need
prompted
with
0
45
satisfactory
two-dimensional
the
plates
walls.
were
these
in
corner
the
This
in
at
wire
be
wire
If
plane.
satisfactory
the
would
the
reached.
be
e.
eliminate
of
measurements
on
to
would
was
plane
layers
wire
re-adjustment
symmetry
0
45
this
(i.
region
symmetry
to
prove
a horizontal
and
the
at
also
would
flow
then
everywhere
carry
orientation
paper
the
horizontal
mild
together.
95
t5 4
45
45
(20
L.E.
/5°
L3
Corner ptate
Front view
View- A
1.5.
2,
Vire
3.
Electrical
4,
Thin
.
Position
electrical
contact
contact
Insulated*
A cotton
the
to
Fig-(3-5)
vrire.
support,
electrical
54
tungsten
pm diameter
wire
contact
pin.
( connected
to
indicator
tie
thread
used to
body of the hot-wire
item
(4)
probe.
of the hot-wire
probe on making
between'pin
and wall.,
96
The
body
of
the
it
to
securing
viewed
through
it.
in
lay
the
centre
having
feeler
electrical
a microscope
when
to
to
the
wall,
was
This
symmetry
plane
and
readings
gauge
was
were
holder
probe
such
calculations
plane,
they
were
In
of
inscribed
on
plane
and
using
between
the
prong
the
for
were
to
plane
the
symmetry
contact
with
the
needle
the
corner,
on
using
of
in
principle'
transverse
one
at
every
be
made,
repeated
the
of
rotation
While
axis.
the
feeler
the
small
and
wire
side
in
required
velocity
the
of
the
of
point
centre
opposite
very
vertical
only
measuring
each
establishing
line
its
nevertheless
measurements
which
by
a Shadowmaster
contact
differences
eliminated
and
subsequently
was
making
the
on
axis
probe
in
length
the
small
were
that
viewing
from
repeated
about
ensure
clearance
knowing
any
to
symmetry,
distance
on
was
position
After
the
vertical
calculated
1.2mm).
by
the
of
measure
point
the
The
when
needle
bonding,
40x.
side
contact
when
flattened
the
and
The
manner
of
one
araldite,
hot-wire.
the
of
handling
of
ease
containing
plane
a quantitative
gauges
wire
by
good
vertical
surface
the
ensure
a magnification
flat
the
by
lead.
point
in
checked
body
electrical
the
to
connected
the
to
for
probe
the
tinned
was
end
flattened
was
pin
line
located
was
on
point
for
plane
the-hair
for
glass
a magnifying
assistance.
Calibration
a standard
it
has
141.
reading
of
in
procedure
been
The
on
in
described
essence
the
a hot-wire
of
the
anemometer
the
is
anemometer
context
detail
in
1
that
corner
references
is
calibration
with
of
on"a
to
more
less
or
flows
and
121,131
compare
manometer
and
the
connected
97
to
and
the
This
is
layer
the
of
sensitivity
limit
is
speed
should
readily
range
for
of
speed
probe
the
wind-tunnel.
the
speeds
near
The
reading
manometer
range
for
tube.
pitot
the
which,
speeds
include
of
from
the
air
particularly
calculated
calibration
in
situated
a wide
work
hot-wire
the
when
suitably
are
over
repeated
lower
the
tube
pitot
boundary
tube
pitot-ýstatic
a standard
is
covered
and
thereby
obtained.
The
the
it
major
of
to
to
respect
(steady)
in
Since
is
to
the
is
developing
who,
experimentally
a constant
the
for
temperature
of
heat
Nu
= hd/k
transferv
is
the
d is
Collis
is
density
air
known
the
induce
be
with
wire
measuring
flow
air
calculated.
this
the
method
method
Williams
and
King
method
profile
the
the
speeds
Poiseuille
[291,
hot-wire
stream
the
wire
izý
-
nI.
BR
Nusse. lt
L281
showed
temperature
a constant
and
such
Instead
by
result
NA+
uw
where
of
work
One
calibration
usage.
earlier
that
can
wire
low
to
used
on
for
of
and
is
the
wind-tunnel
an
tube
speeds.
where
position
frequent
requires
on
velocity
the
at
tube
a
(water
rate
based
tube.
the
of
velocity
for
used
knowing
flow
unsuitable
be
in
centreline
hot-wire
the
probe
The
and
the
tube)
pitot
established.
volume
the
the
of
low
dependent
calibration-method
hot-wire
condition
is
gives
which
very
accurately
however
over
anemometer
sensitivity
greater
potential
range
the
been
this
much
reliable
the
mount
has
flow
in
a
beyond
lying
this
of
existence
its.
hot-wire
the
of
for'measuring
potential
Realization
is
is
tube
pitot
the
advantage
(3.3)
number,.
diameter,
h
is
the
A and
coefficient
B are
98
1
Rw<140
the
value
the
latter
of
in
and
the
0.45.
For
0.06
approximately
rate
per
unit
voltage
is
U is
voltage
a result
This
the
local
U
-1
B*
velocity
is
between
(V/Vo)
of
utility
since
great
2n
lowest
extrapolation
is
valid
is
heat
by
free
transfer
forced
20
work
for
that
out
0C
free
0.125
free
stream
is
to
speed
be
used
experiment
B/
(3.4)
are
a
is
the
is
1281
131
Barclay
5]Lm
diameter
is
This
here-was
1.3
m/s
using
obtained
by
experimental
the
the
always
equation
a least
results.
square
The
method
Equation
have
permissible,
layer
free
profile
stream
the
of
imply
to
the
(3.4)
value
A .4 and
constants
of
at
greater
sp. eeds
range
worked
air
and
boundary
as
and
[41
minimum
speed
that
with
Collis
minimum
such
(3.4).
where
0C
300
at
The
value
of
at
if
the
which
El-Gamal
The
m/s.
therefore
and
fixes
measured.
adequately
the
unimportant
1.25
to
speed
results
comparable
wire,
speed
a reliable
to
basis
is
linear,
being
calibration
to
On the
when
to
the
(3.3)
V0
and
provides
limited
convection.
m/s.
was
n =f 0.45
is
equations
U,
speed
convection
convection
than
transfer
V
where
fluid
and
it
The
Williams
heat
(3.4)
the
of
relation
lower
to
range
speed
U
speeds.
due
at
zero.
extrapolating
much
V2,
enables
measured
to
air
the
to
the
of
means
air
0C
20
as
when
The
an
have
to
m/s.
proportional
2.
(V/V
=
0)A+
where
and
temperature,
hot-wire.
the
rewritten
131.0
to
constant
area
to
for
valid
found
diameter
corresponds
m/s
at
across
be
to
fluid
a
Rw
is
n was
--'44,
w
m wire
JIL
a5
of
range
For
0.02<R
range
(3.3)
Equation
dlv.
Rw=U,,
and
constants
fitting
was
equation
always
99
found
to
See
During
the
the
to
avoid
and
tunnel
any
in
the
involved
test
may
used
method
basic
the
wall
may
will
then
be
applied
by
all
conduction
effect
is
for
to
plane.
be
and
to
be
dependent
of
the
wire
measured
Fig.
(3.7)
in
the
at
the
geometry
and
this
required
several
shows
the
planes
'still
forced
the
that
in
the
windthe
to
anemometer
hot-wire
from
subtracting
this
'true'
the
The
speed.
on
flows.
corner
separately
the
the
of
correction
conduction
noting
arrive
and
of
the
that
parallel
air'
the
to
the
air
of
effect
air
that
determined
be
wheA
local
the
on
is
subsequently
to
reading
higher
air'
independent
by
reading
apparently
'still
the
distances
air'
anemometer
vicinity
effect
out
determined
'still
in
tube)
heat
A correction
case.
therefore
different
The
an
method
The
free-
pitot
boundary
significant.
are
stationary.
and
a solid
on
the
apart
profile
(hot-wire
experimenters
carried
corresponding
found
in
axis
The
uniform
indic'ate
based
can
therefore
reading
be
previous
and
calibration.
the
conduction
a calibration
the
is
actually
the
near
interference,
sensors
the
probe
(. --100-120mm)
a very
close-to
assumption
is
has
two
situated
mutual
is
and
wall
of
hot-wirý
convection
wall
results.
the
distance
small
the
than
reading
a
section
anemometer
tunnel
measured
hot-wire,
always
in
to
velocity
by
the
the
inaccuracy
the
conduction
the
and
of
no
When
were
possibility
separation
The
to
of
tube
wind
stream
on
calibration
pitot-static
of
fit
excellent
(3.6).
Fig.
the
an
give
the
from
value
is
correction
wall
the
in
the
conduction
to
correction
the
symmetry
applied
100
(Y/V, )
2
olts
0.45
1.
0.
0.2.
o. 45
0.4,5
U
Fie.. (3.6)
Typical
calibration
curve
for
a5 Via hot-wire
probe.
CN
x
C: )
LO
cs
.0
0
cs
14
0
%.a
LO
0
13
V-4
4b
C: )
cs
4&
-1
'A-, A
102
in
arriving
4.
The
from
the
for
the
(V-V
line
were
The
remaining
the
wire
be,
the
where
V
the
from
the
four
symmetry
in
indicated
scheme
0
wall
The
negligible.
distances
to
the
from
distances
to
wall
the
of
Chapter
x2
against
plotted
large
according
used
is
in
presented
middle
different
to
refer
and
0
the
to
shown
)/V
sufficiently
heat-conduction
curves
0
wall-to
at
voltage
results
experimental
ratio
voltage
distance
is
the
at
the
figure.
in
steps
the
the
of
preparation
corner
I
in
the
of
zero-pressure
wind-tunnel
the
that
of
plane
from
flow
was
the
corner.
conditions
the
symmetry
adopted
was
laminar
the
laminar
symmetrical
as
satisfactory
the
the
and
criterion
imply
pressure
by
sufficient-distances
velocity
the
is
was
also
already
experimenters
other
flow
This
zero.
be
a profile
such
the
that
was
should
profile
of
criterion
gradient
symmetry
geometric
at
and
established
achievement
principal
used
that
conditions
required
were
the
a Blasius
plane
th e
flow
about
sought
the
and
that
ensure
gradient
The
realized
to
wele
mentioned.
The
at
originating
the
when
This
sharp
or
part.
of
the
of
leading
the
adjusted
was
well
sharp
the
to
give
zero
the
leading
from
is
line
Such
edge.
induce
to
near
and.
stagnation
prone
edge
known
edges
in
non-uniformities
Given
was
leading
the
was
leading
the
corner
gradient.
with
leading-edge
sharp
a situation
one
the
can
of
of
the
two
pressure
experience
caused
on
plane
streamwise
apparently
direction
edge
symmetry
previous
1Aýring
turbulence
under
arise
free-streala
alternatives
by
all
surface
from
minor.
velocity.
can
be
103
applied.
the
The
first
is
to
the
point
corner
obtained
and
pressure
gradient.
flow
direction
that
the
the
surface
of
adopted
here
The
The
The
particular
this
closer
use
conforming
the
and
of
realization
With
an
1.5
m/s
the
required
corner
edge
shown
the
the
flow
flow
ýodel
slat'
speed
of
a series
qualitatively
in
the
the
the
on
a
plates.
on
a cross-secti
justified
on
the
the
of
vicinity
0
3.5
be
corner
which
in
the
local
adverse
in
Fig.
to
leading
the
be
will
has
been
modelled
of
section
out
These
were
obtained
(3.3)).
the
with
(i.
The
e.
a better
nominally,
carried
adverse
edge
theoretically.
was
incidence
caused
which
a whole
0.160,
and
moderate
as
test
conditions.
at
is
disturbances
the
experiments
incidence
at
is
incidence
will
for
air
corner)
edge.
permit
conditions
say,
effect
remove
effect
result
the
leading
thereby
than,
rounded
improve
to
and
Blasius
induced
to
its
turbulence
cor'ner
a flat
its
corner
is
El-Gamal.
required.
the
only
ufficient
with,
one
by
used,
and
of
as
the
corner
shape
Insofar
is
is-that
slat'
ensure
upper
the
the
the
the
on
the
to
edge
consist9of
to
is
leading
of
the
artificially
alternative
edge
a local
modify
and
(rather
plane
the
streamwise
(3.3)
plate
only
of
of
just
only
Fig.
a flat
that
symmetry
to
the
to
upstream
to
approach
of
ground
inlet
is
method
'leading
the
at
is
(favourable)
latter
below
0.37m
flow
everywhere
in
shown
about
of
lies
the
situated
effect
much
vicinity
The
is
edge
incidence
second
corner.
and
plate
leading
The
line
arrangement
wooden
consequent
incidence
geometric
laminar
where
the
stagnation
the
adjust
the
accept
in
to
the
procedure
to
establish
with,
'leading,
c onfiguration
involved
the
104
taking
the
and
The
results
and
(4.63)
the
flow
in
is
flow
the
taken
in
principally
With
throughout
0.805
the
the
m from
traverses
in
leading
taken
were
to
addition
on
two
traverses
vertical
so
Barclay
course
of
random
fluctuations
in
an
symmetry
in
all
at
of
the
the
oscilloscope
but
without
rapid
indicative
of
situation)
lasted
to
normal
it
was
the
seems
very
typical
to
have
high
the
of
for
much
like
time
interval
frequency
few
to
the
smaller
the
between
(i.
12
e.
been
by
noted
in
the
apparently
and
This
manifested
was
trace
the
of
DC and
on
rms
voltmeters,
the
normal
fluctuations
only
(from
(-
10
In
departure.
behaviour
than
plane
a sudden
excursion
seconds
m and
symmetry
evident
the
on
0.51
vertical
has
level
the
This
similar
been
in
readings
turbulence.
being
was
made
station).
also
was
were
five
itself
which
velocity.
rise
the
and
sign
flow
and
the
the
of
by
corner,
M,
plane
streamwise
experiments
in
each
the
edge.
0.343
plane
one
the
leading
m,
side
the
present
occasional
In
El-Gamal
and
the
edge.
on
A peculiarity
Zamir,
0.165
zero
of
measurements
planes
that
confirmed
along
arranged,
but
symmetry
was
the
(4.62)
Figs.
regards
as
symmetry
m from
either
in
doubt
least
positions
0.51
plane
function.
shown
The
in
plane
streamwise
little
at
of
several
corner
are
nature.
plane
corner
the
Blasius
laminar
at
the
the
state,
geometric
measurements
with
leaves
required
and
the
about
different
comparison
gradient
pressure
in
their
and
at
function
Blasius
the
and
symmetry
layers
these
comparing,
the
of
sides
boundary
two-dimensional
distances
both
on
measurements
previously
successive
before.
the
s),
this
return
respect
but
reported
occurrences
The
average
rate
105
of
occurrences
the
but
traverse
to
usual
vertical
behaviour
traverse,
i.
about
more
frequent
common
dependence
rare
event
was
so
that
this
the
was
being
twice
and
mid-length
layer
boundary
the
in
or
be
to
as
it
that
was
phenomenon
free-stream
the
disturbance
leading-edge.
to
in
mild
minutes.
perhaps
at
was
every
15
the
upstream,
of
confined
in
every
of
it
corner
once
than
the
aspect
be
such
than
near
important
to
in
trailing-edge
the
comparatively
once
position
the
of
about
was
downstream
near
appeared
e.
streamwise
mid-length
this
streamwise
An
the
near
the
on
encounter
The
as
dependent
was
any
undetected
there.
increased
The
to
relative
be
due
occurrences
previous
lower
to
[30,
Gold--stein
the
fluctuations
in
of
waS
several
times
main
stream.
This
the
may
possibly
boundary
of
it
these
in
is
tions,
are
likely
tunnel
speeds.
be
to
the
to
The
have
must
indeed
are
that
states
layer
boundary
the
in.
fluctuations
the
in
true
layer
to
to
the
free
the
flow
corner
stream
-I
I
fluctuations
Nevertheless
be
a
attributed
was
within
of
taking
plate,
Gold-stein
stream.
amplitude
ampritude
remarked,
and
case.
present
when
a flat
along
found
the
the
that
fluctuations
ratio
fluctuations
if
free
the
amplitude
flow
was
in
well
may
corner
used
reported
laminar
the
the
although
has
behaviour
random
similar
speeds
3181
p.
in
measurements
experiments
free-stream
disturbances
these
C1
withý900
of
be
flow
been
free
the
source
of
relatively
in
the
were
stream
free
that
expected
large
very
the
vicinity
not
detected.
layer
pronounced
of
the
as
disturbances,
stream
boundary
more
since,
fluctuaat
tan
is
low
106
likely
be
to
low
of
the
to
mean
the
The
is
ratio
is
speed
just
phenomenon
for
boundary
The
speed
the
condition
the
of
more
to
upstream
because
probable
fluctuations
velocity
likely
also
the
and
be
greater
for
reason.
free-stream
in
The
speeds.
air
the
speed
layer
chosen
the
disturb
described
to
be
was
what
layer
results
ance-f
was
in
used
t hickness
boundary
All
steady
unsteadiness
free-stream
same
between
this
of
propagation
least
its
in
presented
flow.
and
in
this
in
factor
to
low
speed
be
the
was
best
disturbance
report
choosing
where
experiments
low
thought
thickness
ree
the
reasons
was
a
were
desirable.
compromise
rate.
obtained
107
CHAPTER 4
RESULTS AND DISCUSSIONS
PreliminarX
The
derived
is
studied
were
from
(2.21),
were
negligible.
rather
intractable
losing
the
feature
non-similar
solve
curvature
These
somewhat
on
(2.25)
variation
of
terms
it
is
the
(2.21)
the
of
the
general
eliminated
equations
character
equations
more
equations
assuming'that
This
of
being
flow
the
of
(2.25).
the
equations
in
To
model
equations
directly
formulation,
explicit
in
embodied
obtained
the
theoretical
I
without
problem.
to
necessary
(2.18a))
(equation
corner
specify
3
i.
(X71e)
e.
L
which
in
(2.18b)
terms
of
the
)
-k
eR
el
)
(
= J'/\
't
U00
-1/p
where
The
(2.21)
To
some
initial
the
difference
(2.25)
becomes
to
a solution
the
requires
paramatically
words
important
and
obtain
plane
(equation
becomes
coordinates
plane
on
clear
at
at
any
step-by-step
equations
the
equation
this
(2.25)
are
R?
'for
any
point,
transverse
from
solution
quantity
(2.25)
equations
of
nature
(2.21)
equations
whereas
of
the
particularly
streamwise
dependent
solution
in
only
In
plane
other
may
be
108
i
directly
obtained
to
corresponding
These
of
on
large
that
remarks
For
interested
are
quite
and
flows,
corner
is
-it
in
developable
to
general
curvature
here,
&
of
plane.
transverse
(2.18a).
th, e value
specifying,
complying
which-we
with
are
equations
primarily
define
to
convenient
surfaces
a parameter
["XC
by
lx"c (ýiRt)-"
=
F7ýC
is
family
(i.
of
e.
covers
differing
all
by'a
only
function
curvature
radiused
corner
in
factor
their
transforms
from
to
according
(3.2),
equations
lies
C
0
and
sharp
corner
desc. ribe
to
this
work
(see
equation
(4.1)
equation
(4.1)
to
give
rx".
)
=Xtanh(t
and
(4.3)
(4.2)
Pt)
(4.4)
U00
in
00
in
is
geometry
xC
and
corners
similar
scale
adopted
tcinh( V4'Rt)
limits
parameter
geometrically
33
where
A one
parameter.
(4.2)
dimensions).
The
which
corner
solutions
corners
physical
the
the
called
?t).
T-C
= -2
the
range
correspond
0
to
respectively
is
the..
where
CO
This
situations.
-
the
flat
from
evident
plate
equation
(4.4).
The
with
of
generality
respect
to
corner
the
present
configurations
theoretical
may
analysis
be
demonstrated
109
by
the
solving
in
shown
the
satisfies
corner
its
(1.2c).
Fig.
3
for
>0
more
is
complex
Section
of
It
geometry.
shape
This
conditions
and
Accordingly
the
geometry
2.4.1,
sufficient
only.
to
corresponding
problem
corner
geometry
still
defining
the
to
convenient
we
may
specify
choose
>0
is
where
the
lengths
scale
I
C
value
of
X
at
the
characterizing
coordinate
system
3=/
xd
and
In
geometry.
may' be
d" as
terms
d
.. d
as
written
(4.6)
is
by
given
(4.7)
or
Therefore
Rrs
addition
to
W
A
in
the
(r/m
r0)
(4.4))).
equations
as
the
of
?,)t anh(4
+
where
with
(4.5)
equation
I"
(4.5)
tL Rt
implying
in
the
the
the
embodies
and
r'
simpler
t Rt/
previous
non-similar.
(as
new
opposed
geometry,
has
to
simpler.
character
r'
and
(3
equation
the
exactly
the
of
is
discontinuous
The curvature
at
shape
of the
complete
requires
cation
(4.5)
functionsequation
and
namely
(4.5)
from
obtained
on setting
equation
(4.5)
Equation
as a special
contains
the
which
represents
mostly
geometry
Equation
(3.1)
by setting
is obtained
discontinuity
is
the
that
of so doing
the
(equation
(3.1))
function
singlS
for
all
x
same
being
case,
3=
d
and
parameters
.
only
1),
(see
significance
a
flow..
in
parameterFor
0 and a specifitwo piecewise
smSoth
for
x
an equation
X
ý
6--d.
and
=-?
(3.1)
case
equation
in this
work.
used
X
/
One consequence
=O=d.
3
disappears
at x=0
and
describes
the
geometry
x
110
define
convenience
(4.8)
Xcp
where
For
want
is
the
of
a better
described
it
angle.
at
description
we
shall
X, ji
previous
a
sharp-profiled
i.
cases,
corners.
e.
sharp-profiled
corners
corner
corner
distinguish
to
corner
sharp
the
call
rX,,,
by
from
corner
radiused
and
radlused- profiled corner
I
I?
t
A
a
cl
ix
2
b
It
to-the
of
that
just
along
of
both
the
the
a sharp
sketch
the
same
corner
of
The
by
were
in
zero
the.
putting
the
the
in
bV
limiting
a
namely,
that
along
words,
with
respect
a and
points
the
for
i'n
Chapter
of
the,
equation
to
c And
equations
the
case
being
situation
(2.25)
equations
the
configuration.
is
this
in
b
point
single
respect
with
in
flow
the
corners
pressure-gradient
terms
is
is
as
sketch)
sharp-profiled
are
described
limit
asymptotic
(see
other
become
in
m=0
re1written
a+
In
equations
As
variable
downstream
far
corresponding
Nevertheless
far
exhibit
flow
which
radiused-profiled
to
so
(2.28).
equations
(2.28)
flow
flow.
AC
made
an
problem
thickness
never
here,
considered
not
and
angle
uniformly
remarks
obtained
limit
coalesce
The
2.7r -
c
the
corner
corner
layer
will
is
corner
boundary
corner
(2.25).
sharp
radiused
the
that
not'e
described
sharp-profiled
approach
the
to
radiused-profiled
the
wfth
case
form
a
ý,
bi
interest
of
geometry
the
to
be
may
clý-
a
",
.1,
to
give
2 equations
new independent
(2.64),
and
the
new
being
ill
dependent
The
les
variab
resulting
(2.66)
with
constants
a and
b must
Of
be
chosen
numerically
to
the
ýmax,
size
ýmax
quantities
The
computing
field
and
were
(2.67).
prior
h
(2.65).
equations
conditions
step-size
five
these
by
solved
boundary
the
must
as
be
to
equations
equations
solution
2
C40L=
defined
and
b
and
are
uniquely
3
through
related
Cvn&v.
l/b.
=
there
is
which
might
Four
no
remain
obvious
analytical
means
suggest
them
computations,
dependent
their
on
to
solution
the
unique
and
in
numerical
suitable
by
trial-and-error
changes
the
similar
All
of
in
any
further
excess
in
used
from
the
a,
values
have
is
the
by
was
Numerical
method
time
parameters
for
100
This
hr.
at
the
and
for
arriving
is
to
seek
for
values
the
least
used
here
360
to
taken
different
was
with
computations
doubtless
introduced
of
IBM
the
on
C. P. U.
the
influence
is
and
(14].
Ghia
done
is
andtmmx
those
experimentation
outset
ýmax
be
hand,
other
bd
may
may
method
2
combination
The
of
the
rational
some
solution
parameters
the
adopted
London.
was
h,
This
that
combination
A
any
by
yet
studied
methods
about
computing
College
h
for
solution.
to
being
scheme.
values
the
problem
of
the
On
choices.
determination
and
solution
but
chosen
their
a
*
independent
quite
at
on
physical
for
that
Indeed
certain,
be
to
arbitrary.
find
we
values.
for
unobtainable
which
be
to
letting
on
where
parameters
experimental
the
(2.64)
equations
so
as
to
a result
University
a satisfactory
reach
treated
corners
large
regard
at
as
to
preclude
the
mesh
this
was
interval
fixed
0.4.
results
for
the
analyti
I
cal
treatment
of
the
112
are
problem
the
with
reýults
Chapter
3.
The
0
315
The
sharp,
data
Sharp
corners
After
some
choice
for
the
scheme
was
made
0t
90
and
in
available
(Fxc=
results
introd
the
cr'iterion
in
the
in
was
each
of
integration.
to
give
the
was
a)
The
=
satisfactory
0.0005
parameter
in
for
all
iterations
successive
in
everywhere
convergence
difference
the
when
from
and
relaxation
used
-
0.5
of
most
domain
the
was
foun.
d
and.
cases
computations.
velocitX
C>0
solution
parallel
than
rapid
Streamwise
r9o
planes
less
most
henceforth
4.2.1
9P
A
above:
mentioned
0
deemed
u,
case
numerical
0.4
was
u es'of
following
the
0.0694)
0.5
val
and
16.0
h
th e
obtained
literature.
the
;
uced
satisfy
tmazv
in
and
theoretical
with
computations
parameters
solution
0
sharp-profiled
14.4
The
are
0,270
225
and
-
co
experimental
to
0j
135
and
in
solution
experimental
themselves
amongst
experimental
theoretical
radiused
theoretical
compared
4.2
45
of
include
00
described
work
for
together
sections
experimental
chosen
angles
These
corners.
are
cases
following
the
the
of
having
corners
in
presented
for
to*the
the
streamwise
symmetry
plane
velocity
is
shown
as
in
seen
r-ig.
in
(4.1).
113
figure
The
smooth
quite
Blasius
the
and
(i.
This
the
in
by
Pearson
(4.2)
In
addition
the
hand
one
plane
and
results
results
generally
in
Fig.
In
u.
velocity
Grossman
layer
than
those
Ghia
[151).
in
profiles
literature
The
have
solved
used
the
same
p.
are
satisfactory,
Ghia's
19
require
[111,
[131
set
results
of
21-23
will
as
is
of
the
).
for
Rubin
of
edge
side
and
expecting
in
often
in
who
the
in
outlined
the
the
all
and
equations
the
the
of
here,
[14,153,
as
(or
here
so
differences
Insofar
simultaneously
streamwise
the
were
this
demonstrated
out
layer
the
and
character
comment
corner
system,
-reasons
evident
the
corner
between
presented
further
cause
the
symmetry
is
into
some
offer
the
results
pointed
to
[15).
of
the
results
been
no
coordinate
pp.
to
shown
solutions
Ghia
and
separation-like
has
full
and
decay
new
The
references
the
(see
the
believed
reasons
of
solutions
with
to
as
do
(4.2)
Fig.
(4.3),
a slower
exhibit
boundary
of
figure,
this
isovels
is
the
agreement
corner
plane
this
effect
Ghia
of
the
existing
from
amongst
the
depicts
which
the
Complete
throughout
is
symmetry
included
differences
those
interest
and
[111
measure
and
the
those
with
is
other.
here
true
(4.3)
the
the
on
obtained
is
to
asymptotic
other
flow
the
profile
which
in
Grossman
and
Blasius
against
the
along
the
are
presently.
profile
of
profiles
primary
and
discussed
profile
Rubin
Of
solution
velocity
Fig.
1131p
and
is
to
manner
0ý0 ).
this
matter
characteristic
reference
t=
the
comment;
a regular
at
e.
streamwise
separately
on
in
between
relationship
The
little
merits
vary
profile
solutions.
is
itself
by
Chapter
1
explanations
present
account
for
agreement
the
114
differences
In
of
Ghia's
remainder.
solution
integration
boundary
conditions
be
expected
be
virtually
for
in
The
decay
fast
precise
value
of
are
applied
would
seem
velocities
and
and
in
pondence
solutions.
as
find
we
results
This
is
u
and
while
speed.
algebraic
conditions
in
the
degree
of
corres-
in
the
two
of
the
algebrai-
variables
w.
and
applied
boundary
high
in
agrees
assessed
from
from
profiles
by
the
case
All
of
support-for
[2,211
Zamir
is
this
numbers
the
are
quite
similarity
how
case
exactly
pre-sent
results.
and
the
where
assumption
results
[41
of
for
different
hypothesis.
streamwise
and
solution
El-Gamal
the
well
experimental
(4.4)
any
El-Gamalls
whereas
Reynolds
Fig.
the
is
course
with
contradict
results
similarity
of
solution
measurements
some
v
(=16)
importance
a
these
importance
of
be
may
velocity
of
for
likely
boundary
the
vorticity,
with
the
will
it.
theoretical
Zamir's
for
less
so
small
look
may
variables
A matter
This
we
wall
increasing
which
have
to
the
effectively
and
w do
and
therefore
must
finite
were
and
plane
solution
at
with
at
the
decaying
cally
v
it
near
applied
velocity
The
symmetry
present
streamwise
The
the
and
they
crossflow
u
direction.
integration
the
of
the
field
the
case
everywhere
were
exponentially
SI
the
present
equivalent
In
ý2
large'
infinity.
in
solution
same.
Ghials
the
on
are
the
the
conditions
whereas
wall
that
in
3
t=
at
the
on
and
infinite
was
everywhere
at
the
with
are
same
may
the
shown,
corner
the'
and
from
even
layer
range
offer
"I
\
115
Zamir's
are
results
a
represent
situation
physical
from
that'modelled
only
speculat&
that
the
what
leading
edge
whose
rise
to
with
such
a. mechanism
to
seems
the
and
obtained
interest
(experimental)
be
the
case.
Zamir's
layer
a better
and
is
observed
flow
a flat
developing
extension
sufficiently
would
it.
this
large
agree
El-Gamal.
If
El-Gamal's
interest.
to
note
intervene
before
be
that
which.
number
considerable
of
unexpected
completely,
have
to
seems
no
belief
El-Gamal
Reynolds
number,
theoretical
some
of
latter
results
theoretical
the
with
in
Reynolds
for
the
for
length'
flow
for
that
apparently
As
measured.
that
suggested
experimentally
an
for
a
me asured
El-Gamal
solution
is
it
correctp
explanation
transiti
turbulent
that
on to
"
development
is
the
complete
the
theoretical
situations
profile
this
number.
an. 'entry
than
is
flow
corner
improvement
greater
profiles
functions
The
exists
much
the
the
between
are
to
increasing
accounts
of
with
are
of
there
this
character
of
consistent
should
a
one
A systematic
very
and
plate
of
and
intrinsic
with
is
which
show
agreement
that
is
Reynolds
not
qualitative
believed
El-Gamal
corner
results
below
source
It
situation
they
be
may
downstream
agreement
profiles
and
lie
a
was
can
situations.
these
the
one
It
measurements
two-dimensional
that
by
agreement
practical
realized.
corner
influence
supported
data
the
view
extraneous
on
for
in
counterpart
curve
best
free
importance
present
obtained.
the
disturbance
a
at
different
amplification
profiles
lowest
in
in
and
that
quite
used
the
behaviour
are
configuration
curious
is
difference.
the
at
and
The
but
prqpagation
gave
experiment
which
caused
They
remarkable..
theoretically
on
disturbance
particularly
is
of
minor
flow
may
in
that
and
be
may
never
116
that
recognized
in
lly
a margina.
the
answer
El-Gamal
from
the
0
Blasius
the
profile
dimensional
He
that
between
I
the
Blasius
solution
to
and
Whatever
to
be,
there
of
the
theoretical
reason
seems
little
theoretical
model
symmetry
the
possibility
a
for
the
with
the
on
effect
of
and
correspondence
profile
This
of
two-
the
plane
plane.
cause
the
was
stream-
non-zero,
difference
the
are
difference
for
cause
realizing
It
is
to
sufficient
the
in
presented
0
90
the
the
questioning
much
more
likely
the
idealized
explain
corner.
following
prove
might
experimentally
for
discussion
results
the
significant
the
analysis.
of
in
is
that
differences,
the
of
precision
some
relevance
seitions.
1"135
The
results
from
experiment.
the
difficulties
This
no
gradient'being
theory
the
in
eliminate
pressure
between
have
would
taken
symmetry
in
contains
also
profile
two-dimensional
profile
considered
the
favourable
corresponding
figure
improvement
experimental
experimental
to
further
a
in
those
with
the
profiles
0
0.3
differences
the
two
shows
plane
(4.5)
Fig.
for
The
symmetry
compared
profiles
concluded
the
raise,
construct
and
plane.
and
[111.
reference
wise
symmetry
To
might
(obtained
incidences)
be
may
gradient.
figure
This
is
corner
a result
possibility
region
coener
the
as
to
(41).
thesis
rectangular
pressure
experimental*data
adverse
for
profiles
small
such'a
two-dimensional
0.85
and
his
a
and
a very
which
his
along
condition
to
question
used
(taken
from
stable
sensitive
extremely
flow
laminar
a
for
the
1359sharp
corner
are
shown
in
117
(4.6)
Figs.
in
planes
the
isovels
less
is
The
problem
1161.
The
the
Blasius
in
for
the
for
to
the
corner
scale
plane
profiles,
plane
profiles
are
have
the
0
a 135
corner
and
these
the
with
data
latter
is
a small'
favourable
have
results
velocity
correction
1311
aware
the
between
effect*of
the
his
the
given
would
is
to
re
(4.9)
important
this
with
and
Desai
the
two-dimensional
some
theory.
and*Mangler
a free
v(l)OC
the
reason.
consistent
1 0.02
(x )
at
them
region
to
in
flow
relate
for
are
of
and
as
great
the
Fig.
It
figu
improve
gradient
as
equivalen't
by
experiment
pressure
in
results
was
differences
two-dimensional
basis
from
between
in
sources
the
a
origin.
gradient
gradient
this
agreement
that
on
distribution
for
in
pressure
calculated
that'
the
included
are
Barclay
in
present
suffers
and
quite
shown,
two
these
the
based'-on
the
solution.
from
corresponding
the
are
Mangler
is
measurements
theoretical
data
same
and
profile
difference
the
the
0
135
Blasius
[7)
symmetry
-and
and
solution
of
obtained
comparing
[161
the
Desai
the
Carrier
of
has
the
and
On
that
for
these
with
Barclays
to
larger
this
and
from
(4.8)
Fig.
of
case.
[31
profiles
(4.7)
influence
available
Barclay
[3,311
together
when
in
symmetry
0
90
the
reference.
and
Barclay
along
plane
defects.
three
the
in
for
0
90
profiles
Fig.
The
plane.
distribution
by
velocity
and
plane
solutions
similar
the
shows
transverse
those
shown
method
same
symmetry
than
symmetry
added
the
the
velocity
are
are
series
(4.6)
theoretical
only
solutions
Fig.
a
marked
corner
again
in
the
on
corner
angle
to
parallel
the
shows
(4.9).
-
Any
measure
The
symmetry
stream
the
quantitative
plane
is
118
but
uncertain
in
that
t. he
justify
The
theory.
corner
layer
'entry
length'
does
weaker,
perhaps
this
at
tentry
the
necessary
to
seem
not
very-much
the
here.
length'
matter
If
of
a
such
of
would
this
at
for
claim
the
arise
the
of
a strong
and
than
experimental
soundness
angle
weaker,
less
resulting
make
results
similarity
is
in
confidence
experimental
concept
the
region
be
not
would
change
two-dimensional
would
profile
the
assuming
be
course
for
than
angle
a
0
90
corner.
r45
F22S F2 r3l
f
ý 7,,-The
symmetry
are
angles
the
new
The
marked
in
categories
(i.
inflection
the
into
flow
the
the
increasingly
0
90
angles
doubt
The
case.
must
as
likely
to
be
to
1800
is
appears
the
the
is
very
in
wall
the
observed
flow
indeed
theoretically
experimentally.
at.
0
this
and
0
pr ofile.
profile
anticipated,
determined
but
to
and
th6re
an
Causes
layer
corner
the
outward
90
below
presence
180
similar
the
precarious
in
the
As
characteristically
of
the
range
angles
point
reduced.
the
the
corner
progressively
corner
with
by
moves
thickening-in
stability
be
is
angle
of
the
at
flow
divides
inflection
no
for
introduced.
the
profile
corner
profiles
are
0f
For
angle
whether
ity
inflection-point.
flow
the
315
and
respectively
rapid
otherwise
The
the
0
of
range
wide
which
stabil
profile)-and
as
in
the
there
Blasius
0
a
characterized
point
reduction
are
an
in
0,270
in
0
180
than
e.
(4.10)
evident.
of
absence
The
Fig.
variation
is
for
profiles
0,225
45
angle
greater
in
shown
corner
and
plane
angles
two
C'o
the
smaller
may
be
profiles...
some
119
For
the
corner
flow
is
(4.11)
angles
transverse
They
clearly
corner
The
exceptional
3150
which
asymptotic
be
may
in
four
only
is
has
its
direct
Its
real
it
an'd
in
handling
of
discussed
computed
and
in,
the
is
in
it
is
not
but
Unity
to
corresponds
the
sense
the
cleai
reaches
step
with
h
size
treatment
numerical
accuracyo
vector
Mangler
1161
on
a result
the
were
of
the
error.
streamwise
in
a defect
by
solutions
in
makes
impracticable.
almost
influence
as
that
the
[71
Carrier
This
was
1.
velocity'
equations
obtained.
is
which
problem
its
crossflow
of
u beyond
velocity
the
for
that
in
crossflow-velocity
lies
the
as
v and w
the
and
is
this
determination
Chapter
from
that
Velocities
crossflow
definition
solution
the
be
this
(4.13a).
and
layer
(4.10)
1.6,
In
isovels
edge.
the
wall
of
(4.13)
for
reason
planes
The
overshoot
fact
may
importance
The
The
in
corner
Fig.
a small
experimental
Desai
in
handling
smallness
component
and
a streamwise
intervals.
mesh
velocity
by
of
the
Crossflow
The
the
withthe
difficulty
4.2.2.
of
1.0.
from
large
very
thinning
exhibits
associated
a distance
Figs.
of
Figs.
cases.
profiles
in
profile
value
and
character
respectively.
shown
form
the
the
0
270
velocity
angles
are
the
show
assumes
the
these
planes
0
225
the
show
for
const.,
in
by
(4.12)
and
than
greater
typified
0
180
equation
vector
(2.17a)
(2.65),
in
components
V and
conjunction
with
once
a converged
w are
the
120
a)
rgo
=
The
oo
most
the
appearance
the
vicinity
Both
in
shown
the
direction
towards
in
or
that
body
be
will
symmetry
plane
the
to
near
wall.
crossflow
is
the
symmetry
plane;
for
outwards
the
vicinity
The
the
complex
shown
in
by
a numerical
Nevertheless
flow
scale
distribution
can
is
the
then
solution
it
a
be
extremely
appreciably
and
hende
towards
experimental
Everywhere
is
flow
corner,
has
eddies
there.
of
viscous
near
the
of
such
corner
the
be
sizes
doubtful
that
eddies
influence
the
streamwise.
bulk
in..
line,
a behaviýur.
structure-will
step
the.
else
0
sharp
a 90
indicative
employing
wall
region
crossflow
fine
the
and-
flow
aeries
this
it
plane.
Stokes
of
'
the
on
the
symmetry
line
is
line
corner
the
points
near
the
121].
Young
such
directed
with
the
analysed
of
exist
eddies
and
the
may
and
wall
points
from
and
plane
two
point
distance
a small
near
of
zero
consistent
corner
'(4.14),
Fig.
the
is
existence
The
the
to
to
who
purposes
first
clearly
at
null
presence
present
is
It
is
the
the
reference
symmetry
region
the
is
Zamir
structure
If
small
the
parallel
of
predicted
Off
of
[20],
Tokuda
other
this
a small
and
the
parallel
visualization
except
For
points'.
and
the
flow
fluid.
convergence
imply
between
vector,
only
and
remarkable.
velocity
'zero
called
is
results
vector
similarity
by
represent
not
in
v
crossflow
does
is
found
also
arrows
a point
The
component
was
velocity
solution
the
the
the
here
crossflow
present
crossflow
crossflow
of
This
from
Ghials
the
line.
Here
a source.
and
the
the
for
the
(4.14).
Fig.
the
of
values
and
divergenc;
a sink
seen
corner
Ghia's
of
or
results
negative
the
of
are
of
of
[141.
Ghia
feature
significant
used
of
of
velocity
the
missed
here.
such
crosscomponent.
121
3
A
in
Fig.
(4.15).
(4.14)
are
feature
The
of
the
in
is
is
results
is
5.
The
abo ut
the
of
these
applied
for
conditions
for
2
large
large
&
they
at
>
little
5)
influenced
(4.17)
Fig.
the
crossflow
symmetry
One
is
plane.
the
profile
2.
the
from
found
from
by
shows
velocity
The
the
results
reversal
is
in
computations
the
flow
in
accord
with
to
be
is
of
planes
two
u is
consequently
the
w component
parallel
to
curious
direction
Ghia's
the
=16)
v.
profiles
have.
=
relatively
are
and
will
value
component
of
in
(C
a finite
crossflow
vector
It
boundary
differences
velocity
Ghia
applied
at
behaviour
the
concerns
20-22).
the
velocity
the
numerical
conditions.,
1 pp.
applied
profiles
certain
these
of
correctly
Where
less
the
of
and
among
for
solution
streamwise
double
This
were
the
Rubin,
amongst
Chapter
were
plane
agreement
boundary
present
infinity.
uncoupled
virtually
of
the
large
than
rather
in
the
[13],
Pearson
principal
(see
at
difference
with
consequences
asymptotic
correctly
that
recalled
but
The
in
v
together
differences
problem.
-ý
profiles
the
par. ticularly
good
its
indeed.
marked
by
(4.15).
is
the
Qualitative
5 are
Fig.
I
component
obtained
beyond
in
a measure
(4.16)
greater
the
of
placing
taking
very
Fig.
quite
v
profiles
as
Ghia'[151.
and
increases
aspects
in
profiles
the
be
be
to
indicated
of
plotted
and
velocity
shown
[111
the
these
=oo
seen
Grossman
as
and
of
profile
corresponding
than
by
ýhe
symmetry
all
be
is
constant
values
to
t
in
0,
planes
negative
small
variation
profiles
The
of
too
extremes
the
The
in
v
illustrated
v
to
sensitivity
of
profiles
Fig.
in
shown
of
series
results
characteristic
the
features.
shown
and
by
is
shape
122
3
&<1.738.
for
The
in
undershooting
analytically
the
has
received
well
as
a
wall
6.
Curve
(2.60))
the
in
=oO
more
results
is
an
we
might
The
shown.
and
the
of
As
in
the
condition
vorticity
force
to
attempt
results
attention.
introduced
parameters
an
the
towards
consuming
reformulation
made
and
expectation
time
of
7
is
uni-directionally,
between
s, everal
a
was
profile
to
deal
the
scheme,
the
comment
asymptote
conflict
great
varying
numerical
at
this
of
for
contrary
case,
resolution
feature
(equation
to
profiles
limiting
of
case
result
exact
expect
this
the
second
boundary
the
3
condition
at
resulted
convergence
For
any
Chapter
the
of
the
it
is
b)
(2.66a,
(2.66c,
equations
approximate
99t2l +
for
device
Another
2
direction
tried
was
configurations
assumed
1
at
'd)
that
u
12.0
and
identically
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apparent
after
iterations.
ý
but
solution
solution
the
corner
solution)
numerical
in
the
and
here.
values
asymptotic
1500
size
briefly
on
diverging
a
about
field
the
outlined
this
to
up
e'xtending
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in
eventually
strongly
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(this
as
a
only
for
well
It
12.
by
confirmed
consequence
4>
their
attain
is
satisfied.
in
considered
of
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now
remains
These
equations
to
treat
equations
to
ýi
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V-lt2*
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123
Analogously
independent
to
the
4,
variable
introduce
we
the
new
variable
Az
71 -a'
where
a' + 41b'
and
with
is
and
99
defined
99 increases
constants.
therefore
by
replaced
indefinitely
t he
bounded
variable
by
çP
The
are
12
for
çO _cf
=
boundary
(P
for
conditions
are
A
-n -61,
t7
4-ax
=
>
(for
12)
*
ý,
ý,
99
=0
-S
, - C
ip
;
*
0'
=
(4.10)
ý
99
71
3 12
where
and
(2.9)
Equations
and
solved
on
orations
approximate
because
to
with
the
the
found
tunately
iterations
to
attempt
to
and
used
whole
converge
were
found
solve
the
it
of
taken
the
in
as
opposed
was
unfor-
More
convergence
problem'by
the
attractive
(2.66),
for
undershoot
from
provides
slowly.
insufficient
and
The
directly
equations
rather
I
of
(2.66).
though
method,
simplicity
relative
transforming
are
This
terms
equations
=-12)
(2.66)
system
equations.
of
It
(at
conditions
in
re-written
simultaneously
interface
it
are
this
3000
than
and
scheme
t4e
was
abandoned.
b)
rl"3
5=
Fig.
direction
oO
(4.18)
pertains
in
corner
the
to
layer.
the
crossflow
It
resembles
velocity
very
vector
closely
124
the
corresponding
but
unlike
is
that
the
on
since
the
we
The
the
profiles
symmetry
of
are
is
zero
0
90
the
the
absence
kind
in
the
rectangular
the
to
for
greater
as
no
flow
is
outward
in
is
angle
altogether
when
similar
in
character
small
but
One
double
(4.20)
plane
respectively.
their
to
significant
in
revers'al
(cf.
the
to
and
the
the
the
w of
Fig.
smaller
plane
disposed
just
diagrýam
the
the
carried
can-be
angle
wall
in
off
0
270
-corner
internal
the
the
is
There
case
and
symmetry
particularly
corners
and
in,
is
2.6
plane.
is
of
comes
the
The
corners.
the,
it
in
0
270
case
a zero
another
length
(4.22)
anti-symmetry
plane
on
a greater
Section
270
inflow
the
over
Fig.
in
00
90
and
symmetry
each
1 on
velocity
crossflow
case
extends
a certain
0
90
corner.
in
this
discussion
find
the
walls
curve
the
of
the
of*,
In
previous
(See
from
the
the
near
to
symmetry
the
this
expectation
parallel
corner
corner.
in
a result
surprise
inward
in
the
found
near
in
(4.19)
the
of
0
270
magnitude.
As
change
with
planes
are
for
shows
line
symmetry
was
and
disappear
corner.
discussion
(4.21)
direction
than
exists
= c<>
Fig.
the
keeping
(4.14)9
0
135
corner.
the
1270
w in
cases
is
over
the
in
Figs.
difference
Otherwise
point
points
and
in
both
for
counterparts.
v
shown
in
profiles
noted
zero
Evidently
result
Fig.
corner,
0
180
is
angle
one
plane.
have
must
the
only
the
and
The
of
case
symmetry
responsible
for
plot
0
90
zero
point
point
crossflow
but
the
on
similarly
This
second-zero
point
interesting.
The
behaviour
125
of
the
of
a
closed
view
Added
to
the
computer
results
diagram.
The
author's
scrap
view
for
i.
e.
this,
a
0
270
the
similar
closed
zero
this
possibility
a
also
shown
might
in
0
90
the
this
might
it
also
be
main
the
be
true
Looking
case?
the
interpreted
be
If
figure.
the
represents
might
scrap
from
forming
view
flow
vortex.
the
in
used
the
in
calculated
this
on
corner,
point
is
not
how
of
suggestive
directions
are
shown
circle
as
flow
points
suggestion.
here,
in
of
is
vicinity,
illustrate
the
region
its
in
To
vortex.
this
of
(4.21))
(Fig.
arrows
the
case
the
at
Fig.
at
again
k,
(4.14)
can
it
be
is
not
placed
planes
not
in
seen
crossflow
are
The
any
interpretation
such
components
in
shown
shows
a
flows
corner
v
(4.22)
Figs.
v-component
internal
the
that
clear
flow.
the
of
respectively.
all
the
on
Profiles
constant
at
(4.23)
overshoot
slight
but
w at
and
and
3
ý=
the
undershoot
3
in
the
w
component
is
still
(see
present
curve
(4.23)).
Fig.
To
here)
the
complete
symmetry
in
large
at
(which
plane
for
remarks
different
is
on
simply
is
angles
the
crossflow
vector,
the
component
v
for
shown
Its profile
ýat
the
w=0
since
purposes
comparison
(4.24).
Fig.
d)
The
shown
in
(4.25)
is
of
a closed
on
noting
resulting
(4.25)
figures
more
and
a similar
in
the
and
increasing
case
even
the
for
(4.26).
that
than
complex
vortex
from
fields
velocity
crossflow
these
The
rq"8
for
of
more
angle
r2,70
cases
in
crossflow
=co
= cO is
pronounced
to
two
0
315
The
.
Fig.
possibility
strengthened
situation
(Fig.
are
(4.26)).
7,
126
The
by
doubt
the
Wall
The
velocity
wise
velocity.
be
page119
..
shear
stress
to
is
in
the
wall
shear
the
forr3l,
accuracy
compared
Consdquently
however
modified
'computational
negligible
stress
that
with
wall
=
terms
the
of
=a
V(1)
2
-; 0C
the
cross-
from
the
stream-
is
assumed
layer
corner
0
COSA)
variables
2
P U. 0
COSA
Denloting
are
e
the
wall
shear
in
prdsented
stress
terms
f
at
ratio
is
the
derivative
second
results
(A. 11)
For'a
function
Blasius
the
this
corner
sharp
becomes
IN)
>0
for
since
The
00
270
and
315
for
are
for
results
for
results
variation
of
(equation-(2.44)).
Tvv,,
case
the
NO)
(V'Q)/C
05
COSA
Twe
added
Zvy,,
ooby
au I
e=
P&
0
-
f (0)
z
&=0,
=-
of
VWC.
where
at
are
the
in
corners
90
corner
The
comparison.
in
close
in
in
shown
0
shear
thi's
of
case.
angles
(4.27)
together
Fig.
as
two
with
by
obtained
sets
The
agreement.
stress
0
0,90
45
of
figure
increase
0,135
results
shows
in
0p
Ac
0
225
the
with
Ghia
cx: o
from
stress
shear
5
by
given
zwc
or
is
evidence
the
contribution
flow
to
additional
regarding
on
mentioned
4.2.3.
this
of
weight
[14,151
0
90
for
the
the
expected
and
near
the
127
the
plane
symmetry
0
180
rises
to
very
shear
stress
large
(i.
for
indeterminately
e.
AC
having
corners
large)
values.
An
is
a numerical
physical
is
[4-,
5
the
or
error
situation
in
noted
in
observed
shear
negative
this
feature
interesting
is
= CIO case.
a correct
uncertain.
figure
the
the
Whether
prediction
of
the
is
not
Nevertheless
it
self-induced
reverse
4
inconceivable
here
exists
be
that
extent
defined
as
the
point
the
wall
the
figure.
the
shear
radiused
the
be
of
used
points
on
to
to
a
the
'grip'
of
Even
shap
in
radius
necessary
for
in
nuierical
the
the
It
that
corner
the
radiused
below
which
coverage
the
corner
described
corner
it
over
becomes
the
for
the
in
special
Section,
2.4.3
there
will
be
part
satisfying
treatment
(ii)
to
and
the
of
some
impracticable
curved
is,
vicinity
the
mesh
solution
This
of
by.
of
number
impossibility
necessitates
the
things
other
variation.
the-immediate
the
for
enough
curvature
is
from
evident
solution
amongst
of
in
is
parameters
a large
part
of
variation
<00)
the
conditioned
important
requirement
the
(0<
The
angle
Ec
provide
on
plane.
corner
to
plane
symmetry
*say.
[32).
conveniently
might
the
values
curved
particularly
this
with
was
requirement
symmetry
from
flow
a situation
such
influence
distance
corners
of
example
corner
stress
corner
get
an
0.99,
choice
a to
and
of
where
The
h
is
the
of
Radiused
4.3.
there
and
The
a situation
(P.
value
provide
simultaneously
.
52
of
the
the
128
have
an
adequate
in
computing
the
course
will
be
the
of
results
values
for
made
the
in
this
Section
work
and
below.
presented
4.1
radiused
in
encountered
out
in
limitations
of
been
are
described
criterion
parameter
has
carried
the
when
the
difficulty
computations
discussed
because
elsewhere
This
time.
Following
of
coverage
the
choice
was
corner
ýMax
= 12.
-
ý It
16.
"ax=
h=
0.4
'N
0.1
4.3.1.
Streamwise
0
velocity
a) 0<
The
.
are
streamwise
in
shown
the
with
corner
1"90
with
in
a
=0
been
already
that
problems
are
flat
the
different
plate
for
As
angle
very
representing
shows
the
methods.
is
too
and
sharp
rate
3
profiles
limiting
The
the
have
which
for
its
to
applications
situations
extreme
The
all
radiused
general
reasons
methods
variation
rapid
of
of
of
change
increases.
to
numerical
variation
sharp
Solutions
the
corner.
the
respect
the
The
in
to
close
the
e.
together
are
extreme*'
increases
the
limited
r9O
of
cases
from
obtained
plane
symmetry
manner.
special
not
each
with
large
ýegular
This
(i.
cases).
special
are
formulation.
corner
of
of
0
represent
by
'1"90
r9o
and
the
values
plate
smooth,
solved
values
other
flat
00
and
different
=oO
and
r9o
values
for
forIgo
profiles
in
profiles
(4.28)
Fig.
rectangular
vary
velocity
the
local
have
in
angle
local
the
When
rj\C
.1
becomes
in
adequately
difficulty
A
with
A
&
Fig.
..
w'ith
&3
(4.29)
in
the
129
important
10
mesh
331
t=0.086_.
planes
0.04
0.086
on
0.51.
first
the
The
and
the
<-
0
range
ý3<z,
two
This
of
be
representation
increment
smallerthis
A
in
(4.3)
Equation
the
of
0.04
4=0.0
and
C=
and
important
an
by
covered
are
between
intuitively,
must,
accurate
which
turned
angle
is
range
influence
the
and
curvature
better.
the
is
tanh
from
it
which
is
exponentially
It
is
Id through
mesh
points
be
representation
r9o
For
size
near
accomplished
the
at
or
whether
but
the
near
coverage
this
not
<
10
For
10
and
the
scheme
used
defect
in
the
any
reasonable
by
the
larger
a
effect
10
and
reducing
ihis
ý11790= co
interpolation
and
and
limit
I\-
to
numerical
is
small
be
be
stationary
no
serious
between
difference
to
to
improved
upper
this
is
had
virtually
theory
the
mesh
persisted.
the
that
the
elsewhere
the
was
suggested
non--Some
noh-convergence
solution
since
vicinity.
given.
of
angle
first-few
were
mesh
constitutes
work
this
reasons
line,
is
It
in
calculations
radiused'corner
present
for
solutions
permit
here.
numerical
result
the
15
the
the
negated
evidently
of
combination
<,
the
practical
symmetry
1"90
and
just
of
the
plane
reason
(4.29).
Fig.
in
of
render
the
increases
value
the
expense
?ý
evident
inaccurate
15
for
to
as
about
improve
3
ý=0
some
problem
for
to
made
also
symmetry
great
presumably
was
effort
so
than
greater
convergent,
the
the
of
is
above
between
will
fixed
This
that
certain
turne
for
Fýc
with
quite
that
seen
3,1t,
enough
adopted
for
the
to
130
intermediate
with
cdses
Regarding
<0.2),
the
the
opposite
lie
characteristically
1ý
value
no
this
(e. g.
results
behaviour
is
the
this
choice
of
the
extreme),
limit
of
for
in
developed
this
0(
[33],
terms,
which
Normally
we
is
this
of
(perhaps
that
taken
infinite)
to
but
S/ro
to
corresponds
the
within,
required
this
practice
=
is
central-
theory
of
R
and
in
reaching
A
the
i
ever
imperfect
is
radius
4 F2
90
should
be
curvature
(reference
---w-00
not
extremely
0.56
which
It
greater.
or
0
be
should
1-9"0
just
small
lie
to
seem
well
that
in
that
conceivable
and
finite
0.2
=
small.
would
is
rather
0(l)
S/rý
that
mean
range.
ratio
the
itself.
1-9-0,
have
from
F90
5 Tr
from
reason
unsuitable
theory
of
which
The
are
be
a may
the
5xR-
below
procedure's
the
where
In
0.15
Apart
suggests
that
and
obtained.
application
is
profiles,
profile
uncertain.
also
small
In
velocity
be
that
very
unsatisfactory.
towards
that
numerical
2
ýmaLz
#ýrAax and
thesis
130).
p.
rather
is
Blasius
could
valid
less
or
the
likelihood
the
be
resulting
reduced
application
requirement
0xx
is
numerical
possibility
the
below
F"90
as
so
present
r
conceivable
where
to
proved
>, 0.15
>1,0
increasingly
extreme
has
solution
0.2
th e range
its
in
error
situations.
practical
for
negligible
the
application
1-90.
the
of
there
is
that
the
theory
no
the
limited
is
the
larger
in
opposite
this
values
that
in
is
treatment
noted
on
to
indication
strong
numerical
behaviour
That
is
lower
side
this
is
some
way
range
GoIn
ý the
However
of
of
profiles
so
any
than
more
fo'r
responsible
corner
parameter.
forjnr
-4
90
>
0.2.
131
One
discussion
present
formulation
It
be
than
is
ground
also
important.
the
so
(see
it
is
flow,
but
is
virtually
in
importance
values
Until
such
times
an
the
with
recommended
is
range
of
is
of
the
it
is
ef;
of
in
shown
AC*
diminish
naturally
the
of
remains
r9o
be
In
to
the
of
the
exclude
difficulties
discussed,
already
of
it
available
truncated
view
for
solution
problematical.
is
values,
practically
values
is
and
=0
44
that
cýo the
complete
provide
<,
corner
radiused
10.
useful
about
(4.28)
Fig.
rgo
4
and
1"90
for
ectively
A
on
large
extreme
190
flow
the
consequently
a
solutions
0
of
approached.
application
0.225
This
be
should
behaviour
answer
r9o
'large
very
terms
will
unequivocal
values.
theoretical
and
behaviour
of
that
On
non-similar
parameter
range
questionable
associated
theory
corner
as
that
lower
dubious
the
the
are
small.
(4.76)).
terms
limits
a greater
suggest
extremes
these
and
of
of
the
r9o
omitted
the
to
near
these
as
appropriate
the
or
hand
the
equation
assume
for
Fig.
that
theoretical
in
terms
other
4.5,
contribute
at
cause
small
the
Section
similar
The
the
final
the
least
at
the
of
exemplified
these
recognized,
on
context
from
that
doubtful
They
the
those
as
thought
results
not
in
omission
such
was
experimental
this
the
terms
may
importance
The
is
of
(2.23).
factor
contributory
possible
but,
more
together
of
ratio
importantly,
the
with
special
presented,
solutions
coverage
The
range.
the
entire
spectrum
values.
to
solution
the
radiused
rXc
and
and
a
discussion
theory
corner
of
t'he
results
is
dependent
will
is
132
normalýy
require
(=
The
constant.
reduced
without
the
of
value
The
if
isovels
in
insight
into
to
(4.30).
is
exactly
The
as
effect
r>-4,,
is"
value
profile
The
(4.31).
the
representative
a
velocity
planes
considerably
one
Such
ý,r
Fig.
in-Fig.
shown
figures
comment
in
=3
different
is
attention
streamwise
190
are
these
the
in
discussion
of
parameter
of
for
shown
loss
confine
corner
variation
is
we
of
r;
the
of
volume
essential
radius
flow
the
considering
-=
with
corresponding
behaviour
expected
and
displayed
require
no
otherwise.
F27o
The
results
The
symmetry
case
12",
70
figure
for
12-,
70=3
effect
velocity
which
to
they
are
4.3.2.
for
those
on
the
the
less
clearly
Crossflow
a)
the
in
' As
comparison.
corresponding
the
solution
is
the
profiles
for
symmetry
sharp
expected
for
profile
velocities
crossflow
are
given
the
behaviour
in
velocity
figures
of
the
(4.35)
streamwise
illustrated
the
to
streamwise
The
plane.
corner
susceptible
'.O<TS',
o < 00
The
included
for
corner).
shows
in
component
similar
the
/Ac.
of
(4.34)
Fig.
below
u
3
C)Ic
The
is
(4.33).
and
velocity
corner)
of
external
(4.32)
streamwise
purpose
well
(radiused
figures
external
the
lies
profile
the
of
(sharp
cýo
in
shown
profile
(4.32)
this
in
are
(Fig.
results
are
(4.10))
changes
in
but
? ýC
.
v and w
field
to
variations
(4.41)..
velocity,
with
Consistent
the
rgo
with
crossflow
3.
133
in
varies
towards
10
the
are
the
the
(=0.225)
are
the
r9o
most
zero
moves
19-0
and
(i.
from
its
to
wall
sharp
from.
increased
the*
its
=7
e.
and
sharp
zero
is
is
situation
or
mentioned
previously,
values
of
the
profiles
the
algebraic
with
is
resemblance
is
in
the
shows
for
between
remarkable.
The
in
this
and
The
is
-
the
of
physical
difficultiesp
the
for
solutions
very
small
decided.
plane
for
of
show
the
r9o
(dotted
the
exceptional
of
profiles
values
clearly
profile
figure
the
behaviour
this
reflection
of
r9o
as
the
of
on
position
Whether
different
the
since
surprising
increasing
with
as
position
position
symmetry
F90
finite
J"90
incl'uded
for
v
decay
= 00 )
as
wall
movement
=0)
wall
Thereafter
corner
warning
be
the
the
rather
obtaining
yet
symmetry
r90=1.0.
r9o
infinity.
advanced
component
crossflow
(
19"0
changes
of
towards
is
(
the
on
uni-directional
plate
that
even
observation
sharp
a correct
cannot
(4.42)
Fig.
change
an
r9o
a
corner
point
here
complex
clear
at
This
to
is
point
wall
its
flat
zero
rather
principal
moves
the
see
is
substantially
apparently
to
it
zero
increased.
expect
point
is
towards
again
further
might
the
reaching
out
is
zero
r"90
for
line
The
of
point
= 0.225
increases,
one
= 0.225
F90
asymptotic
and
marked.
position
The
plane.
r9D
case
situation
symmetry
r9o
for
the
the
as
same
smallest
concerns
Ghia
The
corner.
flow-near
the
with
of
from
limit.
The
it
sharp
virtually
corner)
for
fashion
a regular
1"90
profile
All
beginning
as
well
00
obtained
line)
present
rg"O
the
and
results
in
as
of
systematic
by
the
family
for
finite
the
figure
134
is
the
has
for
result
present
in
discussed
been
4.2.2
Section
I"gr
,ý=
(a).
association
The
This
o0j
Ghia's
with
essence
in
results
discussion
that
of
case
particular
that
was
e
boundary
the
as
by
is
between
that
for
have
already
the
seen
that,
probably
u
the
r9o
the
per. tain
to
the
the
to
these
of
we
is
difference
are
u
Fig.
(4.2)).
used
as
an
example
with
and
v
of
profiles
profiles
velocity
the
of
profiles
corresponding
'
uncoupled,
velocities
crossflow
behaviour
similar
are
Yet
pronounced,
and
(see
='16
r9o
the
v
case
this
and
at
more
where
=3
the
(4.44)
The
is
negligible
of
and
remarkably
1"90
still
variation
w respectively.
is
is
of
(4.43)
Figures
large
of
result
finite
because
2
ý
= 00
Ghia's
for
effect
they
effect
conditions
results
the
results
show
the
on
=00
correspondence
and
boundary
where
on
effect
The
the
at
Any
here
= 00
only
given.
The
results
at
solution
at
results.
1"90
effect
significant
applied
present
pronounced
.the
applying
r"90
the
reasons
most
from
negligible
e.ven
for
be
rA.
be
finite
the
suggests
to
16
so
in
whereas
to
evidently
has
Ghia,
certain
ideally
should
at
placed
this
is
done
was
were
conditions
GO case.
=
12"70
=
(4.45)
Fig.
1270
any
at
planes
the
solution
is
simple
and
there
sign
of
figures
. rend
crossflow
Unlike
pattern
or
showlthe
(4.46)
to
to
parallel
those
and
for
for
are
the
They
00
= 'NO
profiles
plane
are
already
flow
the
v. alues
negative
,e
symmetry
(4.47).
127o
no
Th
vortex.
a closed
r270
for
field
velocity
are
similar
discussed,
of
of
v
shown
in
and
in
general
figures
v
w
135
(4.22)
14'5=
in
(4.23).
and
12z
'35 s
F,
=
The
crossflow
figures
(4.48)
is
here
the
crossflow
together
in
Fig.
4.3.3.
several
Wall
shear
stress
wall
shear
stress
of
values
One
wall
shear
stress
zero
shear
fo. r
by
shear
wall
This
behaviour
that
as
to
values
of
extent
of
the
rather
large
increasing
for
such
1"90
in
is
extent
that
= 0.25,
the
from
the
with
low
wall
at
the
for
the
we
find
extent
wall
sharp
the
shear
corner.
For
slower
variation
effect.
corner
fact
the
symmetry,
i.
appears
)
is
the
the
initial
the
tendency
everywhere
values,
in
for
that
curvature
lower
larger
e.
increasing
reducing
stress
is
extends
approx.
with
non-
exemplified
0
dominates
plane
to
other
with
It
1.0
zero
variation
of
stress.
(>
symmetry
lateral
values
consequence
of
the
of
plane
shear
00
=
the.
of
curvature
the
r9o
keeping
in
nevertheless
from
change
<05
90 ---%
lateril
in
interest
of
The
for
shown
are
(4.53CI)for
of
behaviour
cases,
above
Fig.
case
the
The
=F 27o =3
,
the
the
values
the
values
shear
for
than
stress
distances
ý3,
in
the
are
of
43=0
finite
all
features
confirmation
decreases,
larger
higlier
is
in
shown
Two
*
at
increase
the
rgo
is
in
(4.51).
Fig.
r9o
shown
feature
vortex
of
each
to
are
interesting
3,
=
S
in
v
cases
a closed
corresponding
).
figure.
the
for
(4.5?,
The
1",,
these
most
of
for
those
with
this
field
profiles
plane
The
existence
evident
velocity
symmetry
(4.51).
to
for
field
velocity
curvature
and
p
greater
e. g.
136
in
resulting
stress
shear
is
the
unremarkabl
Apart
shown.
as
the
e,
shear
from
the
to
approach
slower
much
plate
wall
the
behaviour
features
these
stress
flat
varying
r.
with
in
90
an
fashion.
expected
3
ý=0
The
variation
of
the
in
Fig.
*r.,
90
is
also
shown
seen
to
vary
exponentially
that
of
the
The
values
in
initially
the
flat
Thereafter
but
is
in
r45
the
the
corner
for
the
is
the
no
This
the
symmetry
of
stress
before
plane
lower
to
those
than
io
slowly
doubt
the
this
that
physical
important
3
ý=0
n6ar
possibly
stress.
to
corresponds
is
shear
asymptotes
reason
shear
several
phenomenon
where
shear
for
plotted
somewhat
and
negative
the
of
is
effects
and
are
compared
Sharp-profiled
the
shown
results
application
the
streamwise
r,
for
xC =3
additional
here
of
(4.54)
Fig.
and
corner
a radiused
where
the
of
results
in
the
velocity
r,
C%ý
=
an d
NC
.
corner
particular
introduced
of
in
offered
stress
plane
Two
of
1800
representation
case.
angle
shear
symmetry
are
towards
(4.27).
Fig.
A summary
4.4.
is
values
interesting.
small
=3
>
the
a correct
the
is
stress
value
plate
in'teres'ting
near
to
There
Another
feature
XC
3
ý
plate.
situation.
for
curves
rapidly
anything
An
for
value.
with
wall'shear
flat
T=3
Ac
(4.53b).
falling
plate
The
the
for
stress
with
flat
a
the
at
stress
case.
increases
eventually
for
from
Fig.
of
displayed
is
shear
shear
(4.53a).
corner
sharp
wall
wall
primarily
the
theory
of
examples
to
in
illustrate
this
respect.
corner
the
geometry
generality
The
examples
137
chosen
are
is
Each
sketch.
image
the
of
110).
p.
the
first
as
mirror
(for
case
geometry
be
may
corner
whose
constitutes
the
have
of
a-non-
/I
1,
corner,
symmetry
line.
a
used
each
F270,300,3
as. a whole
exhibiting
corner
described
external
geometry
of
sharp
rectangular
symmetrical
and
notation
a semis-infinite
walls
corres-
0.5
=
The
F9O,
0.5
60,3=
adjacent
and
other
F2-70j,
0.5
'2
3()0,3
see
the
the
1-9"0,60,3
to
pond
in
shown
The
other
results
for
the
are
sharp
following
values
numerical
scheme:
will
description.
complementary
The
the
about
and
of
from
computed
radiused
the
corner
same
formulation
with
cases
in
introduced
parameters
ýrnax
the
the
the
= 14.4
emax
'2 16.0
0.4
0.58
Streamwise
4.4.1.
velocity
F96.960,3
= 0.5
a)
The
results
parallel
to
isovels
are
the
of
the
symmetry
shown-in
streamwise
plane
figures
velocity
and
(4.55)
the
and
u in
planes
corresponding
(4.56).
The
most
137
feature
obvious
of
low
the,
of
extent
velocity
The
and
this-is
ratio
is
of
the
in
a
shown
as
reduction
of
corner
and
to
convexity
the
the
this
the-corner
in
results
near
the
form
of
the
0
at
angle
corner
geometry
walls
thickness
of
aspects
the
with
overall'concave
layer
bulging
these
)
stress
shear
wall
consistent
the
of
(approx.
plane
All
layer
the
whilst
and
plane
at
the
of
boundary
the
are
asymptotic
profile
in
shape
boundary
to
rise
giving
convex
smallness
a thick
the
-.y
note
3
ý=9.39
where
3,
ý.
.--
the
of
the
isovels
the
symmetry
of
be
The
corner.
to
greatest
function
a
respect
(4.61)
Fig.
to
appear
region
lead
in
the
in
to
points
near
by
is
undershoot
seen
flow
the
illustrated
profile
plane.
with
bulge
the
Other
region
shear
undershoot
velocity
i. s
results
(4.56).
Fig.
u
the
these
of
the
symmetry
are
shown
thereby
plane
appearance.
'F270.
-30093 = 0.5
b)
The
(4.57)
and
those
work
in
shown
shown
previously
corner
layer
the
flow
Crossflow
4.4.2
Only
are'shown
the
in
in
wall
figures
been
found
most
corners)
thinner
show
a remarkable
this
that
the
than
elsewhere.
the
quite
sensitivity.
shape.
field
velocity
crossflow
the
section
implies
corner
surely
external
is
this.
in
(for
plane
figures
in
presented
are
results
symmetry
This
the
figures
invariably
presented
to
results
many
two
the
has
behaviour.
the
the
of
it
in
example
these
all
results
opposite
of
Of
In
new
this
(4.58).
surprising.
The
for
results
fields
(4.59)
are
and
here
presented
(4.60).
The
negative
and
these
values
138
of
v
or
at
near
plane
absent
corner
examples
are
is
possible
existence
the
the
of
part
corner
Wall
4.4.3.
(4.61).
is
in
(4.61).
Fig.
flat
the
shown
for
the
This
corner
The
a
distribution
for
shear
this
of
shear
Inset
in
by
experimentally
obtained
to
a velocity
of
the
example
being
to
confuse
the
is
experimental
Whether
is
another
results
there
is
matter.
note
and
anything
the
the
flat
to
to
distribution
corresponds
bulging
it
appearance
is
of,
causes
be
previous
plate.
While
to
(4.54).
Fig.
'This
the.
theoretical
shear
overshoot
Young.
correspondence
is
internal
the
stress
shear
here.
profound
in
shown
the
and
a
line
the
different
probably
the
the
for
displayed
discussed
to
a wall
Zamir
which
profile,
interesting,
is
of
behaviour
resemblance
for
than
(4.61)
Fig.
overshooting
this
for
except
values
stress
rise
qualitatively
closer
shown
as
a kind
of
in
(Fig.
wall
distribution
distribution
type
corner
the
Fig.
the
corner
corners,
a rather
in
a rapid
reflection
stress
bears
by
external
in
cases
near
strangeness
is
wall
shear
convex
(4.59)).
external
that
the
appearance
the
the
giving
(Fig.
parameter
followed
distribution
examples
it
is
noting
example
higher
low
on
The
not
the
highlighted
stress
for
sharp
feature
near
0.5
bulging
the
isovels
appearance.
perhaps
vortex
=
the
interesting
most
closed
corner
of
value
A
a
both
produce
plate
wave-like
of
stress
to
here.
in
observed
forl"90,60,13
wall
velocity
(4.58))
symmetry
of
A consequence
streamwise
of
shear
is
This
-
the
important
the
bulges
between
results
given
deduced
from
Zamirls
here.
this,
139
4.5.
The
results
are
British
and
as
in
regions
the
the
This
to
normal
planes
shown,
are
might
the
not
the
0.16
at
positioned
the
the
the
show
symmetry
in
included
for
model
that
and
compared.
directly
further
the
incidence
of
the
turbulence.
agreement
of
the
corner
two
the
corner
will
confusion
any
as
profiles
Blasius
far
justify
those
from
several
by
small
invariably
to
is
is
evident
results
the
Nevertheless
the
may
that
belief
the
to
increases
resulted.
in
be
properly
is
improve
the
in
the
theoretical
This
theory.
attempts
from
profile
experimental
realizes
results
and
away
region
agreement
profile.
to
was
(4.62)
Blasius
the
experimental
since
circumstance
thereafter
thought
with
the
flow
the
adequately
the
as
Figures
Rxcellent
the
situation
experimental
in
profiles
sufficiently
purposes
while
is
of
profiles,
a two-dimensional
figures.
'below'
work
obtained
comparison
overall
agreement
of
instead
avoids
incidence.
adverse
for
these
lie
to
tend
0
the
this
crossing
velocity
coordinate.
results
u -,,4 0.7
about
the
units
the
used
the
different
experimental
plane
ror
prevail.
because
and
only
Throughout
(4.63)
wall
from
been
represents
greatly
arise
in
happens
effectively
in
two-dimensional
the
has
cosA
metric
comparing
in
system
measured
to
For
and
plane
coordinate
alone.
originally
here.
symmetry
coordinate
converted
tabled
are
the
were
subsequently
they
such
profiles
that
lengths
and
units
in
presented
Physical
variables.
be
Results
Experimental
a fortunate
still
adverse
the
onset
140
The
velocity
flow
two-dimensional
to'(4.71)
(4.75).
to
(4.76),
Fig.
variation
1"90
= 0.53,
could
that
positions
values
is
present.
are
the
insensitivity
in
for
unobtainable
than
tation
the
cause
observation
sharp
(4.4),
the
(i.
corner
results
There
EI-Gamal.
the
fact
to
experimentally
rather
close
The
that
also
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that
it
is
distinguish
not
apparent,
is
This
some
some
large
3.2.1.
Section
is
somewhat
experimenone
E41
encountered
and
theory
is
shown
in
this
being
a
for,
in
those
and
obtained
be.
to
an
the
Fig.
work
consideration
altogether
profiles
which
to
Meanwhile
to
be
The
further
introduced
referred
theory,
experiment
without
).
= 00
0
being
theory
experimental
to
e.
in
experiment
r.
the
experimentally
El-Gamal
that
relative
Ego
"4
other
dependence
uncertain.
between
circumstance
0
90
given
is
thing
the
and
and
remain
will
worthmaking
similar
the
desired
which
Unfortunately
mentioned
theory
for
r9o
in
the
clarify
reasons
be
might
grounds.
theory.
between
agreement
with
of
the
and
value,
only
that
0.53
variation
the
to
necessary
poorer
to
with
is
agreement
physical
flow
the
conflict
The
by
on
of
values
were
in
all
The
curve.
C90=
largest
the
others
results
for
profiles
surprising
extent
these
qualitatively
hardly
1"90
from
systematic
the
a single
(4.64)
results,
plane
for
profile
from
by
ed
deduced
of
the
the
figures
in
shown
anticipated
distinguishable
be
be
can
the
Only
.
is
well
symmetry
collected
in
and
figures
in
The
of
plane
presented
isovels
r9o
with
symmetry
I
corresponding
little
show
the
are
region
the
with
(4.72)
in
profiles
easy
which
are
limited
in
in
matter
truth
together.
present
results
are
somewhat
extent
by
141
largeness
the
in
limitation
What
is
by
importance
the
dominant
this
Perhaps
to
we
ro
of
only
is
same
little
must
= 2.5
length
of-approximately
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used
look
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mm would
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m as
is
K, 0.5
approxi-
is
corner
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of
in
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flow
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will
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and
radius
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streamwise
a working
available
them.
on
it
a smaller
for
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of
values
range
having
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from
Nevertheless
experimental
corners
drawn
from
small
flow.
the
corresponding
"c)
T",
radiused
that
suggests
a
This
number.
of
a larger
f. or
different
consideration
have
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to
that
Reynolds
of
is
there
and
conclusions
a
feature
here.
used
results
practice
desirable
for
is
since
situationSin
be
the
the
at
plate
radius
range
flow
the
flat
the
shown
mately,
the
of
the
wind-
142
U-
5.
4.
3.
2.
I-
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0.
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00
(4-1)
Pig.
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Streamwise'velocity
symmetry
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1
parallel
to the
143
Rubin &Grossman[111
[ 131
--Pearson
C) Ghia [151
present theory
I0
if
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Blasius profile
0.
1-
-
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Pig. (4.2)
.II.
-2
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Streamwise
plane
790
velocity
II
6.
.
U in
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1.
symmetry
U
144
Rubin & Grqssmanlll 1
10.
95
.
Ghlatl 51
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8.
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CM
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.
4.
2.
2.
Fig. (4-3)
Streamwise isovels
,
3.
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F9
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0= 00 0
ý
Cos.,?
145
10
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for
Comparison between the solution
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u in the symmetry plane
results
obtained experimental
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146
7.
favourablo
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press.
Practically
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Theory
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Experiments
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r135
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plane
I
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u
u in symmetry
150
Experiments( Barclay
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0.3.
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0
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(4.9)
U
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Comparison
the
symmeiry
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the
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solution
previously
for
u in
obtained
151
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Variation
plaýie)
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symmezry
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152
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Fig-(4-11)
Lgeý-I
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Streamwise
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V
25
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Fig- (4.1*2)--Streamwise- velocity
in'Planes
to the symmetry plaiýe
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parallel
154
I
4.
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2.
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6
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Crossflow
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Crossflow
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163
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10.12.
dir*ection
164
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Curve
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2
3
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4
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14.40
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Crossflow.
parallel
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in planes
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.
to the symmetry planeor 27
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165
2
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4
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Fig. (4.23)
2
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croseflow
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1
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6
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Fig. (4.25)
Crossflow
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t, 1.4f
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Fig.
2.03
s. w
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Crbssflow
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2.5
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1.5
1.0
0.5
0.0
4
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Fig. (4.27)
4-0
b. U
Variation
of wall
8. U
..
I U-u
shear stress
I Z. U
with
corner
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14.0'
angle.
6..
170
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. 4.
39
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1.
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Fig- (4.28) Streamwise ve locity
*
in
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171
50
45
40
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LU
LU
U.
1
30
25
..
20
15
lo
60 .1Fig. (4.29)
.2
-3-
.4
Variation
of the locai anele
yvita distance from tne plane
of symmetry,
.5
6.
172
5.
4s
3,
2.
10
of
U
90
Fig-(4-30)
-.2
.4
.6
ill
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Streamwise velocity
u in planes
to the symmetry plane
4
parallel
i
173
I
0.95
0.8
0.6
0.4
0.2
5.
4.
34
2.
is
0.,
0.
.
Fig. (4-31) Streamwise isovels
¼
5.
1
Is
174
F270=
1.2
1,
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1.8
1.37
0.92
0.6
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00
0
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ite).
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Fig-(4-32)
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Streamwise
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velocity
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1.
3
const.
planes
175
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3.
2.
1.
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M
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40
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Fig. (4-33) Streamwise isovela.
176
9.
8.
7.
C
6.
5',.
4.
I
2o,
is
0.
U
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Fig-(4-34)
Variation
with
corner
.4
of velocity
angle
u pin
sy=etry
plane,
177
CORNERANGLIE=90.00
CORNERPARAMETER=
0.225.
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Crossflow
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178
CORNER
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Crossflow
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181
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183
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.
3
Crossflow
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184
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186
w
.
--
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3
Fig. (4-44)
Cro'ssflow
velocity!,,, w at
const.
planes
187
CMER RNGLE
270.
W
=
CORNER
PRRAMETER
=3.00
p
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Fig. (4-45)
(.
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Cos
Crossfiow
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direction-9
12.21
ce
12.
188
10.
8
c2
ro
v
00
Fig. (4-46)
.4
Crossflow
.81.2
velocity
v in
1.6 -'2
const.
planes.
189
1)
/
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- .2
Fig-(4-47)
0. 0
92
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Cross flow
velocitY
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const.
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planes.
I.
190
CORNER
RNGLF--IrS. 00
3.00
: ORNERPARAMETER--
1-1
f
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Fig. (4.48)
ff
a
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1-51
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191
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CORNER
CORNER
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3
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243
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Fig. (4-49) Crossflow
I-
12."
lot. "
direction
192
CORNER
ANCLE=2ý5.00
CORNER
PARRMETER3.00
f I
f
f
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f
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f
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2.00 '
1648
Fig-(4-50)
8.83
8.0
Crosoflow
18.13
direction
12.23
n. "
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193
315
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fit
(4-51)
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Crossflow
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2.63
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40
8.02
194
cm
°c)
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E-4
246
CHAPTER 5
CONCLUSIONS
The
corner
of
analysis
corner
treatment
cross-section
flow
the
solutions.
in
of
on
By
angle.
has
angles
whether
region
to
the
agreement
excellent
Ghia's
For
radiused
author
However,
agreement
confidence
for
systematic
the
curvature
of
a
to
in
special
the
compare
the
theory
case
of
the
is
already.
solution
In
be
corner
the
on
(at
prescribed
surface.
sharp
corner,
solution
present
the
with
available.
known
to
theory.
by
encouraged
noted
and
best
are
present
the
is
as
can
solutions
the
also
present
well
arguably
other
a
conditions
rectangular
is
in
crossflow
the
the
in
radiused.
side-edges)
which
no
be
as
of
direction.
exist
to
or
of
a transverse
reflected
to
shown
between
exists
variation
is
boundary
the
the
which
implies
sharp
at
corners
with
streamwise
been
and
114,15]
solution
the
known
asymptotic
case
special
in
distribution
the
contrast
corners
behaviour
curvature
reference
In
corner,
range
a special
having
to
this
the
The
as
solution
previously
algebraic
the
potential
without
and
all
the
case
dependent
the
decay
algebraic
corners
the
invariant
to
theoretically.
corner
which
rectangular
latter
is
streamwise
extension
treated
confined
non-similarity
The
far
sharp
is
which
The
sharp
the
a radiused
a useful
so
contains
The
case.
along
offers
configurations
method
new
here
presented
flow
the
of
and
the
the
curvature
smooth
247
between
the
two
reliable
extremes
flat
of
and
plate
sharp
corner.
The
as
crossflow
in
and
complex
cases
some
in
viewed
transverse
streamwise
is
planes,
closed
are
vortices
evident.
A surprising
differences
of
in
them
has
symmetry
of
these
the
being
the
work
the
need.
before
the
of
far
been
has
theory
the
one
the
external
and
conversely
pointed
to
careful
by
radiused
the
The
manually,
previous
experimental
similar
to
However
there
theory,
is
experimentally
to
represent
is
from
the
is
emphasizes
The
comparison.
due
to
the
work
required,
is
a
experimentations
of
nature
to
This
primarily
workers,
velocities
realizing
extensive
What
All
in
and
consuming
the
corner.
streamwise
differences.
is
in
corresponding
the
data
to
pertain
obtained
analysis.
conclusions
done
sets
predicted.
attempts
data
experimental
qualitatively
for
time
a
difficulty
the
experimental
been
for
the
the
of
Two
than
that
further
tedious.
extremely
of
walls
experimental
vindicate
drawing-firm
shortage
vicinity
its
sets
relation
responsible
for
when
with$
on
marked
corners
results
are
although
which
extent
dealt
the
theory.
higher
likelihood
model
cases
experimental
a similar
entirely
large
has
the
therefore,
some
so
and
significantly
not
in
curvature
the
solutions,
results
do
test
present
show
theoretical
sharp
obtained
to
used
of
two
the
corner.
corners
course
is
solution
walls
two
independently
been
sharp
the
concave
internal
Three
for
curved
In
small
the
of
results
slightly
has
the
have
the
plane.
corner
for
feature
an
automatic
the
výhich
and
this
248
traversing
much
permit
would
data
and
more
the
of
to" be
to
results
This
system.
processing
traverses
exhaustive
treatment
statistical
perhaps
and
collection
made
with
improve
confidence.
There
to
be
dealt
important
and
flows
suggested
approach
is
no
Of
with.
are
those
flows
probably
where
that
of
shortage
corner
amenable
with
such
developed
studies
here.
are
to
the
treatment
significant.
conveniently
remaining
problems
non-zero
is
compressibility
flow
most
gradient
pressure
It
pursued
is
by
the
249
APPENDIXýA
On the
by Desai
derived
To
in
derive
the
details
need
corner
boundary
by
be
give.
u-
AA
Desai
as
boundary
Mangler
derived
by
Desai
necessary
the
of
Mangler
are
0
(Ala)
JOA -
A,
jifg
U U93
-
jlfpj
-
U tlt, 2
(Alb)
(Alc)
-J-o
yl 6&
4- 3
43
and
u,
YJ.
-3=
the
equations
goUP2+
9912
conditions
only
governing
ýPA,,+
Y33At
[161
and Mangler
and
The
n.
layer
conditions
asymptotic
correct
formulation
the
boundary
asymptotic
ri
I'A
j Q9
=
.;,,
(Ald)
with
L2.
y33
-2
(uý)
Yij
where
(i,
-r
23
il
+
i-tia-2
functions
J=1,2,3)are
'r21
J33
=
-
(3V)3
only,
and
of
Vi2i
jo
iý
ozi
JJ3
'0
W)
%
ý
and
a comma
before
a subscript
with
respect
to
The
variables
i
denotes
differentiation
dependent
non-dimensional
are:
u
Aa
streamwise
velocity,
quantity-related
covariZnt
component
to
of
the
streamwise
vorticity,
250
ý10 F
and
x2
and
ILr(2)
U
to
related
are
x3
v(2)
components
velocity
physical
the
contra-
in
velocity
components
3
ý
directions
respectively.
variant
The
to
related
quantities
jqD
Crj2 U+U
YJ
and
and
the
along
v(3)
by
U00 4X
C", 40-x
A2)
[di 332
LI
V(3)
151 (i,
where
1
O(R
to
v 13
from
which
-0
Desai
crossflow
value.
UCO
=0
ý
=W
23
-
ý
Mangler
and
As
only.
and
of
velocity
that
assumed
a
approaches
vector
as
whence
0
and
w, 3
(A2)
equations
U,
v
"3:
functions
Chapter
the
constant
+
J=1,2,3)are
in
remarked
3
2L
+
FU
23 0
in
resulted
LL
(A3)
for
and
33
43
to
according
The
their
coordinate
;
If
components
of
and
the
and
coordinates
continuity
ý
X is
Mang ler's
ýý
v".
+
;;
are
the
ý,
and
33
then
vorticity,
in
definition,
vorticity
C'ý 1)
physical
23
t;
and
the
along
as
obtained
non-dimensional
streamwise
notation,
be
may
vector
the
(1.2).
equation
conditions
velocity
equation
and
system,
boundary
correct
follows.
Desai
00
the
terms
of
are
+
02 ý
(6ý
w
FV)#
IY3
3
ýý
0..
30
(A4)
(A5)
251
3
For
tz
-ý
0
iTTP,
30
*3
which,
when
in
substituted
32
63
43
e-.,.
gives
since
.Z0
i. e.
(A4)
equation
const.
const.
00
2
From
in
'at
0
symmetry
the
limit
0
and
since
3
(53 710
(or
we
have
infinite)
then
for'all
e-+.
00
Similarly,
from
the
definition
continuity
(A6)
a'
00
In
Desai
and
from
which
Mangler's
(A6)
coordinate
leads
33
Cii'Z-G93
system
to
0
W-,Z
or
Lv"
;=
e--3100
It
can
be
shown
const.
=
that
/3o
these
say.
are
results
to
equivalent
(A7)
and
when
boundary
substituted
conditions
in
on
(A2)
equation
(p
and
Y/
as
lead
3
4
to
the
required
252
a
(A8)
00
2
or
The
the
boundary
asymptotic
flow
variables
conditions
be
can
_6136;
0
as
from
obtained
-jb-cwo
the
for
asymptotic
e
series
for
series
is
large
by
used
to'that
similar
Desai
used
and
Pal
by
(This
Mangler.
Rubin
and
1101).
2 -n
n =o
00
M+l
=Z
(ef
"=o
3
Ancý)
(A9)
n+i
99
Z
4/4
>
0r
n=O
Upon
solutions
(A9)
series
substituting
were
for
obtained
in
the
zeroth-order
II=_. 13
IB it
ý2
B(ý
where
is
2B811
to
subject
B(O)
+ cos.
boundary
=
prime
2
BB"
den*oting
following
equations
.2
RO function
18
41
is
governed
(AlO)
2COSTA
and
by
=0
conditions
0
Bf(0)
B1( oo ) --othe
Blasius
the
the
L 13
1.,
?
COS2ý
(Al)
equations
2
co9 13
differentiation
with
respect
to
3
1ý..
253
Consistent
first
the
order
boundary
boundary
with
equations
have
conditions,
for
Ul
the
solution
was
by
shown
So
far
It
sound.
the
is
boundary
22
4
-. ).. oo
the
that
Desai
their
with
to
corresponds
01
with
large
affect
Mangler's
and
is
A1
and
together
x
is
Mangler
and
of
(A3)
Desai
(A8)),
=0
Desai
solution
conditions
In
adversely.
by
the
when
(or
Mangler.
and
analysis
V
and
43)
u,(63)
as
(A3)
conditions
perfectly
that
sought
the
assumption
the
solution
development
the
equation
f or 99, is
05
11
11
99
2
A
V
L
43,
8
Bl
4_.
Sina
CoS
12
the
and
boundary
associated
atz B0
+f
'S
conditions
(A 11)
are
32
#f
CC5ß ti,
In
from
contrast,
boundary
in
(A12)
incorporates
e__*_oo
at
flow
implies
of
equation
the
,
the
two
terms.
underlined
whereas
with
The
assumption
the
correct
(A13)
conditions
of
sets
condition
region.
same
(A13)
term
is
conditions
in
equation
two-dimensional
strictly
underlined
the
boundary
underlined
of
in
three-dimensionality
(All)
the
=0
II
between
difference
lies
(A9)
series
9
so(o)
the
and
are
conditions
The
(AS)
equations
(A12)
0
in
The
(A13)
solution
254
slnA
cosI3
(ef-L
57 =-
B)
tcoso-242-4g
ginACOSA
+B
eo
where
is
from
obtained
Ji
(A2)
the
and
order
zeroth
solutions
2 COSA
A similar
result
be
can
A
for
obtained
from
the
relation
Cos
The
by
obtained
It
can
R. H. S.
of
Pal
an
.
shown
for
2,
however
arbitrary
Rubin
angle.
redefining
(A14)
to
ý
the
suit
is
the
result
according
to
variables
on
[101
for
2.6
more
a rectangular
the
cartesian
3.
is
taken
the
essentially
Section
is
by
where
is
result
and
Ch.
(A14)
be
system
this
(see
Mangler
and
equations
coordinate
by
Desai
(A14)
equation
(A12).
conditions
that
in
term
underlined
the
for
general
to
normal
same
as
corresponding
similar
in
it
that
wall,
obtained
variables
example).
that
the
is
Equation
derived
(AM
for
255
APPENDIX
Soluti6n
Equation
(2.49)
ov I+f2+2c
'r
.
-I-
the
and
of
R
(2.49)
equation
is
S*w
yje-,
1B)
boundary
appropriate
conditions
are
2
=0
Yll
00;
The*solution
integral
of
(1B)
equation
Integrating
form.
take
can
(1B)
equation
l a 7.
a
closed
once,
we
cc*
get
C-1
+
(213)
C
is
where
boundary
integration
an
found
constant.
(DO)
conditions
(00)
(00
by
the
using
to'be
=0
ci.=_1e
On
integrating
again
21
(L I-
C
C
where
found
2
to
is
(2B)
equation
2#
it
*'tf-CS
C)f
f)f
integration
another
becomes
constant
.0*2flý
and
4-
is
Ct
(3B)
similarly
be
c2
=
Carrying
out
the
following
steps
IN
i)
replace
in
the.
f
by
L. H. S.
(from
of
equation
equation
(3B),
(2.44))
256
ii)
(iii)
divide
throughout
integrate
iv)
between
limits
the
o.f
0
and
4,
and
0
use
the
f
by
A/
result
is
2
3
f v2
c6
f"
(4B)
0
With
the
Blasius
numerically
by
0.02
in
function
using'Simpson's
f
being
rule
known
with
Yj
is
evaluated
a step-size
of
257
APPENDIX C
Solution
Numerical
Equations
finite
the
domain
of
(2.66)
difference
mesh
size
the
For
example,
are
approximated
(ý-Uý
C)
.*=ý
7
the
grid
(U
2hi,
j+l,
(ui+ltj
Wh
(U,
(U
.1
h2i,
(U
(i,
-
uitj-l)
-
ui-lpj)
shown
in
i
Fig.
2u
j+l
i+l,
1.
2u
j
of
the
Lk
7ý)i3 = Wh
these
and
j
index
(4u,
not
+u
iv j
points
are
used
variables.
u
of
to
i-ltj+l
coordinates
three-point
expresq
+1
,j
u 10 J+2
j
-
3u,,
ui+2,
approximations
differentiation.
at
the
e. g.
3u
denote
as
end-difference
derivatives
domain,
j
j-i
i-ltj)
-
+,,
i-i,
ijj-l
and
integration
(4u,
+u
i
order
used
(finite-difference)
does
each
derivatives
the
j-l
the
Second
are
2h
In
j
and
approximations
boundaries
j)
i+l,
h2
suffixes
mesh
dependent
point
J+l
+,,
4h
The
grids
square
approximations
the
The
method.
interior
of
by
solution
by
i-2
)id
At
difference
derivatives
at
numerical
into
1).
(2.66)
iterative
divided
Fig.
central
express
2
6Uuu+u
is
ksee
h
to
suited
Gauss-Seidel
integrati6n
of
second-order
to
are
Equations
of
J)"
a comma
Using
such
separating
258
approximations
(i,
j)
(2.66)
equations
interior
at
grid
point
become
2
2
)IJ
Ui-i,
-i+
[(c (P u..
+
t[CeD
- Fle-)
C 21ij
[S 4,, IA,
2 41 j. j,
41)U, ] i
-2
i,
4
,
*1
I
C),
-93) jjX
+ C2(
(,, a)
E)ji
--
)f
ei x.
,(+
c
'
em,
r_
-2
19i 1,j) +
[S41
"24
]I.
[(cq
l'
je
-x
1
lip- Z) ýj
i e
43C4L
mv/,
je+
+
cp"t)
+c(
4-
eccs42fe-9)u
-,
C51ii
ij +E
TC,
A2
Y.
4,
33ý
t,
+9xU
P3
s
-6 -eu,, )0
ex ( 4,ttP4
ý-
F
-1-
lid
(2C)
-
1
c
k2
fi
ý12L( cpi
+
+Cpjfj-J)
L
+
-, -t
cpi
S, 3)
-Z
C
4-
-c
?,
+
ý-,
(
ylý
+
)cp,
-%c'-%
4
I+4,2
4
cp"
C,
+
3
4ý ey
(C S13
2cC,
513),
+
3)
t7
(ýý,
-2
+
C4
S13)"s Y;
ý4
ALA
(3C)
259
tj
c
cp,.
"t
-,
2
.93,
c
)
-X
-), c S.,3ý
Ut2
V/12
41. (2
C
-
43C,
3) t4,
4
-
cy
ý3
In
brackets
are
The
of
the
GC)
equations
evaluated
dependent
(2.67),
on
the
Uisl
=*
Ria
Conditions
a)
Conditions
itj
-.
conditionsp
**gM-1
ýý)Jjqj
the
symmetry
(4u
2, 'j
a
sharp
-
U3, j
plane,
corner,
L--(u
3 ý,
J'(4 P'tij, (P3.
sj
I
(0ý
*
i=l
Fig.
1, j+l
x-
cpj.,
boundaries
the
J=l
i=1,2,
for
5
j).
at
yy
Izon
square
boundary
the
using
wall,
::: : -,
gh
(ii)
evaluated
(4C)
follows:
Conditions
= Cý4
in
(i,
point
are
by
0
quantities
grid
variables
as
c+033) U
all
the
at
domain
integration
equations
(4C)
5*lipj
k-",(Cplpj#.
L
(2.2bl:
u lpj-l
260
for
Conditions
b)
a
boundary
The
ýorner:
radiused
for
conditions
u
at
and
strictly
are
U14 = cp,4=
When
these
were
upon
changing
b,
Consequently
after
lead
to
to
symmetry
conditions,
(3C)
and
the
found
plane
applying
after
of
the
become
(U
Ul'!
+ U,
+
+(C'3
-03
At
and
used
2
2.1
41
+
h2
I"
q9 are
and
u
results.
(10)
equations
extent.
directly
conditions
satisfactory
constants
undesirable
of
equations
these
symmetry
an
the
of
values
to
the
influence
to
the
full
the
applying
found
(2.64),
equation
form,
finite-difference
their
conditions
solution
a and
in
used
[(C
E
S,31
+
CP -A)
(Ul
qI'j
Uz'i
+
h2.
At+2C
+ tA
+
U'.2 +c
W
S"
*1
-
,jý
The
boundary
are
as
beforev
+D
becoming
and
with
e,,
ct. t
ID"j,
conditions
i.
*,
fAp-:
) =vY,i
on
e,
G*N
it.
and
at
4 =0
CP'
'j
261
Conditions
c)
U1, J
for
the
(4 U2,
j
-
example
U3, J)
in
shown
-
'2-(Ul,
(2.2a)
Fig.
UlIJ-1)
j+l
o
0
cp3Pj)
Tjj(*4-fPj
cp
- S(SP-Ljj+L -
Ivi-A)
+2k se ul'i
[34,
0
(iii)
Conditions
in
the
J=N
region,
potential
10.
-3-»N--01
(4
c
cpi, m
2
li,
-A
N
3
Conditions
at
the
side-edge
of
the
i=M
layer,
corner
ce)
f
El
csIs ce)
ICO
c
E)X,i
3
K144
The
guess
for
s
=
iteration
U,
um
Igs
=x
ýs
C6
process
commences
A6
-f
P and I& which
conditions$
e.
1
11
++
017Z
Io
+sct,
with
satisfies
e))
-I
the
following
all
initial
boundary
262
After
iteration,
each
dependent
factor
and
is
n
condition
was
all
the
integration
The
corner
the
any
the
of
to
according
n-1
(P
number
or
V.
W is
the
iterations.
of
when
satisfactory
n-1
x
iIi
<
dependent
relaxation
The
solution
following
the
0.0005
variables
at
every
grid
point
in
domain.
choice
values
of
attained
xn-.,
itj
for
be
value
relaxed
D
u,
the
to
considered
new
+ (1-&>)x
u3 xn
X represents
where
the
is
variables
xnj
iiIiiIj
is
the
of
of
the
a and
configuration
size
of
the
step
in
the
Chapter
integration
size
are
4.
given
domain
for
each
and
263
I
iv
Co
ýq
0
to 0
%.
ý-
1
U
-72
::-ý
a
El
0
9
Cd
It
4J
w
41
0
ps
04
+31
0
4D
40
H
P4
264
REFERENCES
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