measurements of Hoheisel et al. [1984], o_= 0° ........ 136. (a) Comparison of predicted and measured pressure distribution for the Wortmann. FX 63-137 airfoil.
A COMPUTATIONALLY
EFFICIENT
LAMINAR
SEPARATION
FINAL
MODELLING BUBBLES
REPORT for
NASA
Paolo
Dini
Department The
Grant
and
Mark
D. Maughmer
of Aerospace
Pennsylvania 233 Hammond University
NAG-I-778
Park,
July
Engineering
State University Building PA
1990
16802
OF
ABSTRACT
In predicting the aerodynamic characteristics of airfoils operating at low Reynolds numbers,it is often important to accountfor the effects of laminar (transitional) separationbubbles. Previous approachesto the modelling of this viscous phenomenonrange from fast but sometimesunreliable empirical correlations for the length of the bubble and the associatedincreasein momentum thickness, to more accurate but significantly slower displacement-thicknessiteration methods employing inverse boundary-layer formulations in the separatedregions. Sincethe penalty in computational time associatedwith the more general methods is unacceptable for airfoil designapplications, useof an accurateyet computationally efficient model is highly desirable. To this end, a semi-empiricalbubble model has been developed and incorporated into the Eppler and Somersairfoil design and analysis program. The generality and the efficiency have been achievedby successfully approximating the local viscous/inviscid interaction, the transition location, and the turbulent reattachment processwithin the framework of an integral boundary-layer method. ---%
Comparisons surements numbers
of th%i_redicted for several from
2,000,000
airfoils down
aerodynamic show
excellent
to 100,000.
characteristics
with
and
agreement
consistent
experimental
mea-
for Reynolds
PRECEDING
PAGE
BLANK
NOT
FILMED iv
TABLE
OF
CONTENTS
LIST
OF
FIGURES
.........................
vi
LIST
OF
SYMBOLS
.........................
xi
Chapter 1
2
INTRODUCTION
.
°
o
o
.
°
....
°
..........
1
Previous
Models
7
A Model
for Airfoil
LOCAL
The THE
.........................
Local
..................
9 13
.......................
Independent
13 15
Parameters
PART
OF
Distribution
THE
Controlling BUBBLE
the
Bubble
.............
.......................
of the
TRANSITION Criteria
Approximate
Envelope
Parameter
Governing Closure
40 43
........................
43
d'
Method
Flowchart
47 52
..................
57
....................... PART
OF
THE
62 BUBBLE
........................
Equations ....................... .............................
Supplementary
Functions
29
39
Criterion .................... .........................
Error
TURBULENT
Scaling
...............
.........................
Eppler's Transition The e n Method Drela's
Singularity
..........................
Empirical
22
33
Separation
Angle
......
29
.............................
Separation
Model
.
..........................
LAMINAR
Removal
THE
.
......................
Philosophy
Results
Closure
The
Design
PARAMETERS
Pressure
5
.
1 3
Early
4
.
Background and Definitions .................... Problem Statement ........................
Modelling
3
•
.....................
.........................
............
67 67 72 73 77 89
v
6
w
RESULTS
............................
NACA
663-018
NASA
NLF(1)-I015
Airfoil
Eppler NACA
E387 Airfoil 65-213 Airfoil
....................... ......................
Wortmann 7
Suggestions APPENDIX: REFERENCES
w
63-137
CONCLUSION Summary
r
FX
Airfoil
93 ......................
Airfoil
....................
Conclusions
for Future MODEL
Work
SUMMARY
..........................
98 108 135
..................
......................... and
93
140 141
...................
141
...................
142
.................
144 150
vi LIST
OF
FIGURES
Page_ Fig. I-I
Schematic
of bubble
and
Fig. 2-1
Prediction
of bubble
pressure
and Fig.
2-2
Stratford
the
the
[1989],
using (52)7
(H32)n 2-3
Fig.
2-4
Comparison
3-1
3-2
Green's
Fig.
3-3
Comparison
Fig.
3-4
3-5
Green
4-1
=
Fig.
Fig.
4-2
4-3
of the and
Comparison
model
with
and
Mueller
[1988]
turbulent
boundary-layer
Transition
at laminar
turbulent
Reynolds
number.
. 20
boundary-layer Transition
at 24
part
of the
parameter
bubble
as function
............
velocity
shape-factor
dissipation
profile
correlation
32 .........
for the
34
Falkner-Skan
laminar
coefficient
laminar laminar
correlations
for the
Eppler
region
singularity
shear
boundary-layer inside
for the 37
demonstrating
the
.............
41
derived
transition
layer
.............
lengths E387
correlation
...............
separation
Goldstein
development
36
profiles
of two empirically
Eppler
those
recovery,
McGhee
Drela
reversed
of the
layer
Drela
laminar
Green
separated
Modified
Stratford
are from
number.
and
Comparison
shear
data
bubble
with
.....................
Comparison
for the
U_r = Us,
gradient
of the
of the
obtained
23
in the
profiles
of the
airfoil
compared
19
.....................
Falkner-Skan Close-up
program
transition
and
Eppler
two-parameter
removal Fig.
(2.4),
Eppler
pressure
Fig.
E387
of Horton .....
by Schmidt
at an intermediate
recovery
Eppler
5
[1988]
the
of the
Pressure
McGhee
incorporating
Comparison
separation
from
Somers
Experimental
laminar
Comparison
Cp,
.........................
and
and
distribution
inviscid Eq.
of the
velocity
program
separation
of Gaster's
Fig.
from
are for the
at a low Reynolds
methods
Fig.
the
= 1.51.
methods
Data
Eppler
characteristics:
on the
distribution.
characteristics
original
obtained
Fig.
recoveries.
Aerodynamic with
of its effect
predicted airfoil
45 by the
two
at R = 300,000
development the
correlations
bubble
plot ...........
showing
.....
46
the 49
vii u
Fig.
4-4
Comparison
of Eppler's
amplification
surface
Fig.
4-5
Neutral
curves
Fig.
4-6
Amplification
Fig.
4-7
Growth
curves
5-i
Normalized
Fig.
5-2
Comparison
shape of the
and
the
present
5-3
Summary
of bubble
Fig.
5-4
Example
of iteration
distribution
Fig.
5-6
shape
recovery
pressure
of bubble
Fig.
5-8
Variation transition
of supplemetary .........................
NACA 6-2
NACA Fig.
6-3
those
interactive the
........
65 ......
70
of Eppler, 78 parameters with
obtained
in governing
......
81
inviscid
equations
affecting
with with
laminar
length
bubble
height
. 88 at 90
and
airfoil,
laminar
measured
a = 0 °. (b)
a = 0 °. Data
are
thickness
original
obtained
with
program
XFOIL
Viscous
of the Eppler the
Pressure
by Gault
present of Drela, Tunnel,
analysis [1955] the
at laminar by Gault
NASA and
. 91
distribution
inside
transition are
bubbles
for the
growth
a = 0 °. Data
the
separation pressure
development
by assuming
airfoil,
Low-Turbulence
velocity
87
parameters
characteristics with
on bubble
at transition
of momentum
Aerodynamic with
airfoil,
663-018
obtained
64
86
boundary-layer
663-018
that
.....
function
to merge
for modelling
663-018
and
Comparison with
method
scaling
number
of predicted
NACA
summary
factor
83
height
of procedure
Comparison
for the
63
....................
Variation
(a)
........
development
correlation
and
terms
5-7
6-1
. 51
........................
the
Fig.
shape
on undershoot
of different
Flowchart
of the
flow
...............
geometry
Comparison
5-9
factor
recovery
function
Fig.
Fig.
factor
Effect of decreasing Reynolds distribution ........................
Fig.
shape
envelope
turbulent
line,
for separated
boundary-layer
erroneous factor
Fig.
5-5
values
the
Dashed
criteria
of the
for two
with
profiles.
attempted
for two values
to Drela's
Fig.
Fig.
of the
of n in a non-similar
compared
criterion
for self-similar
for H12 > 4, is one
Drela,
transition
..... bubble
separation.
[1955]
.....
NLF(1)-1015
Somers bubble
R = 500, 000. NASA
LaRC,
96
airfoil
program model
94
compared
and Data 1987
with are
the from ....
99
viii w
Fig.
6-4
(a)
Comparison
for the
NASA
is XFOIL
Tunnel,
summary
and
compared airfoil, 6-5
(a)
with
line is XFOIL
Viscous
summary
analysis
m
Fig.
6-7
NLF(1)-1015
the
Low-Turbulence
the
those
(a)
[1988].
development 6-9
(a)
Comparison
for the from
Eppler
McGhee
boundary-layer a = 1.5 ° Fig.
6-10
(a)
Comparison
for the from
et al.
Comparison
et al.
Fig.
with
Eppler McGhee
boundary-layer a = 7.4 °
the
and
upper
(b)
present
XFOIL
of Drela, .................
Eppler
of predicted
and
summary airfoil,
.........
106
airfoil
obtained
compared
with
and
the
with Data
are
[1988 I. (b) Viscous
pressure
distribution
are
McGhee
from
and
boundary-layer
a = -2 °
measured
et al.
pressure
Experimental
......
analysis
110
distribution
a -- 2 °. Data
Eppler
are
summary
E387
and
airfoil, 112
and
measured
E387
airfoil.
Experimental
et al.
[1988].
(b)
development ........................
1987.
development
model
measured
for the
LaRC,
109
airfoil.
of predicted
a = 6 °. Data
R = 300,000.
E387
development ........................
distribution
NASA
E387
a = -2 °. Data E387
104
pressure
program
bubble
at
..........
Eppler
[1988]
analysis
prediction
a = 4.8 °
XFOIL
and
boundary-layer
Tunnel,
Somers
are from 1987.
boundary-layer
of the
airfoil,
surface
Experimental
and
and
Data
LaRC,
airfoil
airfoil,
Viscous
for the
with
distribution a = 2 °.
NASA
Pressure
of predicted E387
pressure
at a = 2 °.
Tunnel,
summary
Eppler
program
Eppler
NLF(1)-1015
experimental
airfoil.
characteristics
McGhee
for the
measured
and measured
NLF(1)-1015
original
interactive
6-8
analysis
obtained
from
of predicted
NASA
NASA
102
NLF(1)-1015
NASA
Aerodynamic with
Fig.
NASA
Comparison
for the
analysis
development
for the
at a = 1° compared
Viscous
line
Low-Turbulence
(b) Viscous
prediction
(b)
are from
the
1987.
airfoil,
Pressure
(b)
from
and
Low-Turbulence
for the
a = -5 °. Dot-dashed
prediction
the
(a)
distribution
.....................
NLF(1)-1015
a = 2 ° for the
pressure
boundary-layer
of predicted
NASA
development
6-6
are
LaRC,
surface
XFOIL
Comparison
Dot-dashed
Fig.
Data
upper
measured
airfoil,
NASA
a = -5 °
for the
and
NLF(1)-1015
prediction.
Pressure
Fig.
of predicted
Viscous
for the
pressure
distribution
a = 9 °. Data analysis
Eppler
E387
summary
are and
airfoil, 114
ix
Fig.
6-11 Aerodynamic with
Fig.
6-12
the
characteristics
original
obtained
with
Data
are from
McGhee
Comparison
for the from
McGhee
6-13
(a) from
6-14
(a) from
Fig.
6-15
for the
of predicted
for the
and
Experimental
et al.
[1988].
(b) Viscous for the
from
McGhee
boundary-layer a = -1.6 ° Fig. 6-17 (a)
Comparison
for the from
Eppler
McGhee
boundary-layer a = -0.5 °
distribution
analysis
and
et al.
[1988].
(b) Viscous
are and
airfoil,
pressure
distribution
o_ = 7 °. Data analysis
Eppler
are
summary
E387
and
airfoil, 122
and
Eppler
Somers
present
of Drela,
[1988]
.................
of predicted
and
and
with Data
with the are 124
airfoil.
Experimental
et al.
[1988].
(b)
of predicted
obtained
compared
model
measured Viscous
for the
airfoil
R = 100,000.
E387
development ........................
E387
program
bubble
XFOIL
pressure
distribution
a = -2 °. Data analysis
Eppler
are
summary
E387
and
airfoil, 125
and
measured
E387
airfoil.
Experimental
et al.
[1988].
(b)
development ........................
distribution
summary
E387
measured
Experimental
et al.
airfoil,
pressure
Eppler
airfoil.
the
and
a = 0 °. Data
E387
with
are
120
of predicted
Eppler
E387
measured
airfoil.
program
Eppler
pressure
Eppler
E387
original
Comparison
117
118
of the
(a)
.............
analysis summary
characteristics
McGhee
R = 200,000.
(b) Viscous
Aerodynamic
from
with
[1988].
development ........................
obtained
model,
measured
for the
interactive
compared
et al.
Eppler
the
obtained
a = -2 °. Data
McGhee
those
6-16
and
boundary-layer development = 6° .........................
with
Fig.
[1988]
airfoil
Experimental
Comparison
for the
program
bubble
et al.
E387
airfoil.
Eppler
McGhee
Somers
present
development ........................
boundary-layer = -0.5 ° Fig.
the
Eppler
E387
Comparison
for the
and
of predicted
Eppler
boundary-layer a = -1.8 ° Fig.
Eppler
those
(a)
of the
Viscous
for the
pressure
distribution
a = 0 °. Data analysis
Eppler
E387
summary
are and
airfoil, 127
w
Fig.
6-18
(a)
Comparison
for the
m
prediction (b)
of predicted
Eppler
E387
Viscous
analysis
6-19
(a) from
Fig.
6-20
MeGhee
and
Experimental
et al.
[1988].
(b) Viscous
Experimental
et al.
[1988].
(b) Viscous
boundary-layer development c_ = 6 ° .........................
for the
(a)
and
Hoheisel
upper
surface
airfoil,
are
of Hoheisel
Comparison
for the from
summary Wortmann Doppler [1988],
[1984].
FX
Brendel and FX
and
upper 63-137
Velocimeter a = 7°
are
summary
E387
and
airfoil,
distribution
e = 7 °. Data analysis
Eppler
measured
Experimental
to the et al.
63-137 Mueller
surface airfoil,
and
are
summary
E387
and
airfoil,
pressure analysis
[1984],
o_ = 0 °
measured
........
136 distribution
a = 7 °. Data
(b) Viscous
of Brendel
65-213
Velocimeter
pressure
a = 5 ° compared
and
NACA
Experimental
boundary-layer
are
summary
for the Doppler
[1988].
distribution
c_ = 0 °. Data
Laser
airfoil.
measurements
......................
distribution
pressure
development
of predicted
Wortmann
at 129
analysis
(b) Viscous
boundary-layer
measurements (a)
airfoil.
o_ = -1 ° compared
prediction
133
of predicted et al.
boundary-layer
pressure
measured
airfoil.
6,5-213
[1988].
c_ = 5 °. Data
Eppler
E387
NACA
XFOIL
is XFOIL
131
of predicted
Comparison
et al.
surface
measured
airfoil.
Eppler
distribution line
.............
and
from
6-22
with
airfoil
(a)
McGhee
McGhee
E387
Comparison
pressure
upper
for the
for the
Fig.
and
boundary-layer development cr = 4 ° .........................
from
6-21
E387
of predicted
Eppler
for the
Fig.
from
summary
Eppler
Comparison
for the
are
at oL = 1 ° compared
o_ = 2 ° for the
measured
_ = -5 °. Dot-dashed
at a = 2 °. Data
development
Fig.
and
airfoil,
analysis
development to the and
for the Laser
Mueller 138
xi LIST
b
width
of separated
C
airfoil
chord
Cd
section
cf
skin-friction
Ct
section
lift coefficient
Crn
section
moment
f
disturbance
h
distance
h_r
height
gl
laminar
g2
turbulent
length
Y/2
exponent
of velocity
FZ
linear
stability
p
static
pressure
r
decay
rate
drag
OF
shear
SYMBOLS
layer
coefficient coefficient
coefficient
frequency
length
quarter-chord
point
in Hertz
of separated of dividing
about
shear
layer
streamline
of the
from
airfoil
surface
at transition
bubble
of the
bubble
distribution
theory
power
amplification
of dissipation
function
downstream
coordinate
It
streamwise
velocity
inside
the
tt 1
streamwise
velocity
above
turbulent
shear
layer
112
streamwise
velocity
below
turbulent
shear
layer
tt I
streamwise
fluctuating
u,
friction
v I
l
turbulent normal amplitude
velocity
velocity, Reynolds fluctuating
_0 stress velocity
of fluctuating
velocity
stagnation boundary
flows
In (_-0)
streamwise
--pttlv
the
for Falkner-Skan
factor,
coefficient from
law
point layer
of reattachment
xii
X
distance
along
airfoil
y
normal
Ao
disturbance
A1
amplitude
of turbulent
H32-distribution
upstream
A2
amplitude
of turbulent
Hs2-distribution
downstream
B
van
CD
dissipation
distance
chord
from
the
amplitude
Ingen's
surface
at neutral
separation
angle
of reattaehment of reattachment
= 17.5 o,, - pu'v')_-_ydy -_y
coefficient,
C_-
maximum
DU
inviscid
velocity
F
reduced
frequency,
G
amplitude
Gt
turbulent
H12
boundary-layer
shape
factor,
H3:
boundary-layer
shape
factor,
P
Gaster's
R
chord
turbulent
shear
stress
decrease
coefficient,
as/71
_
modified
pressure
wake shape
gradient
parameter,
length
Reynolds
number,
scaling
Tu
turbulence
U
velocity
U_
freestrearn
Ot
angle
thickness
factor
Reynolds
to match
in Green's
profiles
62
u_
SF
....
parameter
number,
momentum
U 2/
oe
function
Reynolds
R62
_-
U=
of Coles's
laminar
_,5cr p
AS
Ks_t_ v number,
curvature
u_Ax8 V
of H32-distribution
at reattachment
intensity at the
Falkner-Skan
edge
of the
boundary
layer/inviscid
velocity
velocity
of attack
wavenumber
#
constant, ] 3 fo(# 1/2ou
coefficient,
pressure
stability
relative
to the
of sinusoidal laminar
chord
line
disturbance
pressure
gradient
parameter,
_s
dU ds
w °,,
Xlll
t
Clauser
turbulent
pressure
gradient
7
separation
6
boundary-layer
thickness
61
boundary-layer
displacement
62
boundary-layer
momentum
63
boundary-layer
kinetic
6
spreading
of turbulent
angle
angle
nondimensionaI #
viscosity
1/
kinematic
parameter,
thickness,
energy
velocity
fo (1- _) dy fo _ (1- _) _y
thickness,
U
thickness, shear
ratio
1-
layer
of turbulent
shear
layer,
of air viscosity
of air
dimensionless
streamwise
(FS
Falkner-Skan
characteristic
P
density
Tw
wall
CO*
radian
A
turbulent
defect
h_
Horton's
universal
coordinate
inside
boundary-layer
the
bubble
thickness
of air
shear
stress
frequency
of sinusoidal displacement reattachment
Subscripts: OO
undisturbed
flow
condition
/
the
imaginary
T
the
real
S
laminar
7¢
turbulent
reattachment
T
transition
point
TE
trailing
part
part separation
edge
point point
dy
disturbance thickness, parameter,
fo
u-'_ dY U.
(_)_ dU
It 13VU2
Chapter I INTRODUCTION
This thesis is concernedwith a particular aspectof the overall task of estimating the aerodynamic performance of airfoils. Before a more precise statement of the problem can be made, it is necessaryto describeits physical setting and to define some important terms. Background The can
prediction
be obtained
air over speed
it.
aerodynamic
by letting
The
speeds
of sound As the
of the
the
airfoil
of interest
of the
number,
Definitions
performance remain
here
air to justify
Reynolds
and
stationary
are usually
while
airfoil
analyzing
sufficiently
an assumption
defined
of a two-dimensional the
smaller
than
of incompressible
flow of the
local
flow.
as U_
R-
c
(1.1) p,
falls
below
approximately
of momentum
before
outer
flow
verse
pressure
is, the it.
cannot
separation such
that
surface.
This
range.
transition
amount
pocket separation
Due
reach
to the
the
separates
is usually
a small
on the
can only
layer
and
Laminar ber
gradient
cases,
= 4 x l0 s, the
transition
readily
boundary
In such
R
leaving
of fluid
bubbles ability
near
a thin
region
free
shear
by reattachment remains
the
turbulent
layer the
surface, velocity
of reversed layer
the
separation
are observed shear
layer
run
imposed
of the
flow.
flow
adThat
underneath of laminar
boundary shear
out
of the
the
downstream
between
may
momentum
as a turbulent
trapped
a laminar
on airfoils
of the
Since
by a negative
in the
is called
boundary
occurs. fluid
be balanced
followed
of fluid
surface
stagnant
occurs
laminar
layer
layer, and
the
bubble. over
a large
Reynolds
num-
to reattach
to the
surface
2
of a streamlined imately first
two
orders
observed.
observed At
shape
this value,
sition
and
number and
Unlike
in Reynolds state,
large
may
downstream
extreme
forty
attention
may
depending bubbles
below
larger ease progress
been
one
near
on their has
recent
Piloted
form
the
length a fortunate
(although
they
of measuring toward
their
between/{
that
burst
it causes
layer
to reattach
at
high
is generally development
Vehicles)
has
shifted
million.
In this
one.
the
range,
causing
thickness. Since
The the
appear
to be very
pressure
or flow
full understanding
trailing
to a deedge.
in lift
numbers
upper
In
and
near
surface
is ascribed
bubbles
the
and
= 40,000
a the
at high
in this
case
observed
by
gradient.
separation
bursting
on the
Reynolds
due
decrease
Reynolds
present
of tran-
critical
of the
a large
at high
peak
has
upstream
bubbles
told-chord
downstream
is usually
a bubble
is
22 = 300,000.
but
pressure
years,
below
the
is
coet_cient
of attack
shear
leading-edge
in drag
immediately
reattach
laminar
jump
is approxseparation
airfoils,
suction
since
since
In more
to values
bubbles
years
laminar
approximately
also occur
adverse
at which
For
and
of the
of the
in 1933,
(Remotely
terest
great
first
of the
inability
values
stall.
aircraft
greater
The
Jones
abrupt
will usually
Bursting
to the
of the
number
number
results.
of a cylinder,
in drag.
of attack.
most
case
Reynolds
a large
reattach
angle
bubble
angles
Melville
and
a long
edge,
In the
cannot
it is called
increase
leading
tude
the
cylinder,
separation
on thickness
that
is decreased
bubble
laminar
critical
than
a circular
number
the
massive
200,000.
crease this
in fact,
the
smaller
flow over
Reynolds
depends
as an airfoil,
of magnitude
In the
as the
such
Reynolds believed of small
Reynolds due
to the
significant shift latter similar
variables
of this
were
from are
first
numbers
to be responsible remotely
range
delayed
leading-edge
in profile
an order
has
it possible This
drag,
to mid-chord
flow
phenomenon.
of in-
transition,
in internal made
for
controlled
number
increases
usually
received
of magni-
structure),
the
to make
understanding,
2-2
however,
is still
Whether crease
between
conditions,
occurs
by
have
to refer
to them
the
means the
immediate
bubbles
due
the viscous
however,
to its
to establish
gradient,
be obtained
ited
sufficient
bursting
in pressure
interaction not
not
a reliable
to a decrease
resulting and
the
traditionally
been
as "weakly
purpose
separation over
bubbles
airfoils
called
From
the
cid
point
study
of view
bubble,
distribution
recirculating
flow between
thinness sumed important
a model
lift
these
limiting
bubble
is lim-
coefficients.
Such
it may
for weakly
incompressible, for the
of boundary-layer
therefore,
In addition,
not
is to develop
account
inviseid
interaction.
are
from
can-
be more
accurate
Statement
interaction
to the
pressure
to assume
although
properties
by the
at low to moderate "short,"
in the
accurately
viscous/inviscid
modification
interacting
induced
strong
two increase
interacting
dimensional, in airfoil
laminar
viscous
profile
flow
drag
that
them. the
strong
can
section
in-
interacting."
developing
that
accompanies
and
of this
and
or to an
by a global
Away
interaction
Problem The
regions
approximations.
especially
number
is characterized
inviscid
viscous/inviscid
criterion.
in Reynolds
flowfield
of simple
vicinity,
bursting
that
the
of the
to be valid signals
but
a separation is elliptic the
very
however,
the
everywhere. are
first
Rather
communicated
on
distribution
is concerned
Given
bubble,
pressure
only
flowfield
applicable.
is made
is such
when
theory,
and slow
basis
the
the
speeds
upstream
point,
the
through
the
A weakly
global
invis-
a strong
local
of a region it seems
reeirculating
flow are
recirculating the
of
of
natural
approximations
approximations
within
weak
magnitude flow.
by
boundary-layer
of the
boundary-layer than
on
description
a reattachment that
viscous
characterized
with
between
of the
as its effect
is still
and
the
distinction
by the
insofar
confronted
the
interaction
and
the
usually
as-
flow,
in fact, with
the
outer
flow.
A parabolic
laminar
separation
at least
a local
and,
bubbles
w
the
with
In order
algorithm.
an integral
must
This
computational
is therefore to the
has
been
efficiency,
the
adequate outer
for analyzing
elliptic
employed
flow through
in the
present
boundary-layer
model
development
is
method.
to determine
development
method
as long as it is coupled
interaction
to maximize
calculated
boundary-layer
the
increase
be calculated
in drag
through
caused
the different
by a bubble, regions
the
shear
of the bubble
layer
as shown
-i
in Fig.
1-1.
by
laminar
the
The
formation boundary
boundary-layer design
tracked
and
work. the
in the
aration
and
bubble
transition
the
occurs
downstream
corresponding
turbulent
to the
is possible, does
not Rather
it is more
highly
usually
smoothly especially
reach
from
than
describing
informative
reaches below
its fully
developed
with
inviscid
before
the
the
effect
to explain
why
be
flow,
which
near
the
the
numbers, trailing of the such
that edge bubble
an effect
occurs
bubble
point,
which
inviscid
for
must
this
causes
the
leads
a plateau
to laminar turbulent
to the
pressure
1966],
relaxing the
accuracy
corresponding
[Green,
non-equilibrium,
sufficient
integral
development
T¢. As an additional point
at low Reynolds
equilibrium
region
Using
figure,
layer
separation
From
recovery
at point
shear
in the
surface.
with
the points
region.
"undershoots"
layer
the
8, shown
airfoil
to turbulent
figure,
of a reattachment
boundary
merges
layer
free
between
a pressure
shear
of reattachment
laminar
transition
the
be determined the
in the
at point
from
can
separated, from
of the
encompasses
turbulent
point
distribution
end
is initiated
separating
As shown
velocity
of the
gion
this
Once
T, predicted.
to form
the
layer
methods,
airfoil
point
of a bubble
seppart
reattaehment
recovery
always
the
velocity
distribution
boundary
layer
downstream
distribution. state
and
velocity the of the
Eventually, the
the
undershoot
distribution. turbulent
of
re-
Clearly
boundary
it
layer
airfoil.
on the
pressure
is observed.
This
distribution, can
be done
7" \ \
U
%, \
Uoo
\ \ \ % \ \
INVISCID
/
\ \
\\
DIVIDING STREAMLINEZERO-VELOCITY LINE
,-
Fig.
1-1
/__
_
-_I-
Schematic
of bubble
4 _2 g---
and
of its effect
on the
Velocity
distribution
6
by invoking
two
conservation tion
of linear
of mechanical
layer
the
stresses,
as this
Bernoulli's
rise
the
boundary
layer
negative
As the
results.
This
runs
the
airfoil
pressure
In the new
the
imposed
gradient
pressure
pressure
flow,
this
kinetic of kinetic
be recovered. law
such
by the
same
induces
inviscid
flow
matches
what
the
boundary
layer
is strictly
development,
the
it to
As soon
outer
a lesser at
that,
allowing
gradient.
felt
shear
is a scalar
of reverse
pressure
the
out
can
no diificulty
it, in turn,
boundary
of energy
it has adverse
the by
runs
is a vector
decreases
growth
of static
layer
the
conserva-
available
hand,
reverse
decreases,
a bounded
distribution
other
effectively
it the
and
the
dissipated
for
no more
of the
inside
conservation
inviscid
presence
being
boundary
of momentum,
surface,
limit,
the
flow,
While
by trading
since
on
energy
incompressible equation.
quantity,
out
the
however, of the
flow.
Once
of separation,
to balance
happens,
reverse
layer.
momentum,
of mechanical
unevenly
achieved
is a positive-definite of linear
a modification
and
is still
boundary
downstream
energy
gradient.
is simply
pressure
Conservation
become
energy
conservation
In an inviscid
static
shortly
the
momentum.
is continually
energy
once
the
pressure
inside
and
laws,
total
energy
law
conservation
flow as pressure
growth
of
constant
pressure
energy
equation
allows. While
the
momentum
on what
is "handed
transfer
of momentum
tion in the bubble the
reverse
present
down"
by the
by the
brings
in a laminar
the
Reynolds
stresses
momentum
of the
flow to be accelerated
reattachment
point.
as a rapid
boundary-layer
This
upstream
loss
and
context
of an ifltegral
method,
growth
in momentum
thickness
in the
the
flow
measured
momentum part lost
in this
part
the
This
to be recovered
of the
of the
of transi-
wall.
is observed,
by the
cross-stream
downstream
flow near pressure
turbulent
momentum
appear
outer
near-inviscid
of outer
growth
that
efficient
dependent
by
allows by the
definition,
bubble.
Within
the
flow results
in the
large
bubble
and
can
result
7
in a significant
increase
in airfoil
drag.
Previous
Previous empirical quent
attempts
correlations
:
for the
reattachment
Navier-Stokes =
over
empirical
models
proposed.
Von
tangent
sition
location
this
problem
Doenhoff airfoil
by
Reynolds
The angle
Crabtree
of the
bubble
distance
of the
than
shear
many
others
a coefficient
and
subse-
the
the
years,
streamline
several were
and
the
present
of pressure
with
transition
along
determined
using
model
followed rise
tran-
formed
of 15 ° measured
have
long
is straight
number,
is then
layer
by
characteristics
separation
that
crude
He determined
to reattaehment
Interestingly,
proposed
dividing
between
turbulent
streamline.
more [1957]
distance
early
bubble
transition
the
the
very
turbulent
caused
the
point.
streamline.
years.
In
separation
dividing
configuration
studied.
for particular
a constant
separation
is partly
at the
the
initial
been
from
Reynolds-averaged
surface
and
dividing
of the
that
at separation
of the
laminar
assumed
velocity
direction
has
range
[1938]
assuming
spreading
from
of approaches
correlations
the
a constant
phenomenon
length
variety
or empirical
to the
flow
to full simulations This
which
this
transition
behavior
equations.
time-span
and
at modelling
Models
resembles
in the
in the
from
next
turbulent
the this fifty part
m,
equal
to
- ;r This
coefficient
happen are
for values
discussed Gaster
He
has
been
of cr > 0.35.
extensively [1967]
characterizes
used
These
in review
investigated the
mainly
bubble
by
to monitor early
papers
bubble the
(1.2)
results, by Ward
bursting values
with
of the
bubble
bursting,
measurements, [1963]
and
decreasing momentum
which and
Tani
would
hypotheses
[1964].
Reynolds thickness
number. Reynolds
8
w
number
at separation,
Us( 2)s
(R6.)s = and
a pressure
gradient
(1.3)
LI
parameter,
p_
Au
(i.4)
/ks
where ent
the
velocity
between
instance
He found
that
bursting
a fundamental
and
Reynolds
can be traced
Although
in Gaster's
separation
decreasing
evolution eters.
the
gradient
role
modelled
in the
present
governing
mated
inside
the
bubble.
many
years.
distribution growth are
along
inability
not
approximately
which
1970,
detailed
McDonald
[1975]
is matched
study,
these
same
[1980,
the
stream
of the
bubble
and
Gault's
[1955]
measurements
of the
steady
in computer by
the
inviscid
to an acceptable
equations
degree
layer
methods being
an
prompted
a
flow field.
solution
flow.
shear
shortcoming
have
finite-difference
outer
for
pressure
These
technology
Navier-Stokes
boundary-layer
The
2.
problem
to the
process.
bubble
approxi-
in Chapter
bubble
approximated.
of the
plot. play
coarsely
detail
the
main
on this
a semi-empirical
approximation
their
bubble
parameters
are
studied
adequately
simulations
two
for
two param-
line
proposed
transition
or general,
a local
with
1986]
to the
same
in more
gradi-
conditions,
in these
equations
by a good
ellipticity
viscous
[1967]
will be discussed
advances
developed
with
in this
is not
local
variations the
Horton
inviscid
corresponding
along
and
accurate
for the
measure
mean
varying
always
et al.
however,
sufficiently
of more
axes
is distinguished
bubble,
With
of points
boundary-layer
region
to the
occur
model
Ingen
bubble
to account
Since
and
the
Van
refers points.
a locus
model.
integral This
model
in the
usually
number
and
Their
would
is not
the
[1975]
number,
bursting
where
Ingen
reattachment
on a plot whose
model
Van
parameter
methods. for the upstream
Their
of accuracy
Briley
bubble and
predictions but
region downmatch
comparisons
with
airfoil
teracting
drag method
distribution for
the
bubble
direction,
flow
inside
one
in the and
their the
[Gault,
1955], are
not
[1979]
method
interacts
of the
inefficient
there
one
in the
well
the
degree
outer
model
they
and
advantage
Giles
employ
of time,
for the
[1987],
of such
methods
about
next
characteristics
effect
of the
Cebeci
are
predicted
bubble
report
on
and
airfoil
Kwon
and
boundary-layer
limited
present
to the
in their
[1989],
the
vortices:
in between,
presented
local
by
time,
the are
generalmuch
too
routinely.
have
is that
for the
a finite-difference
approaches
[1989],
a source
different
data
in-
method
details
three
one
for the
by
accounting
methods
and,
that
flourished:
boundary-layer
Drela
the
where
flow
an
boundary-layer
bubble
[1983],
These
to be used
period
and
the
and
Schimke
flow.
of complexity
formulation
cases
and
inviscid
to be
part,
developed
interesting
appear
While
formulations,
the
rather
experimental
number
Cebeci
with
same
sense. the
outer
By properly
turbulent
with
similar
turbulence
termediate
Drela
instance,
eomputationally
Over
integral
For
opposite
[1984]
a finite-difference
to resolve
Reynolds
follow
and
is able
method
Carter
of the
layer.
considered.
Pletcher
and
boundary
compare low
Davis
bubble
and
in the
given.
a perturbation
the
part,
rotating
not
on
bubble.
laminar
method
ity
based
flowfield
by this
drag
are
representing
flow
wall
data
be
interactive
development.
and
they
can
Gleyzes
et al.
combine
a relative
considered methods
Crimi [1983]
and
of an that
use
Reeves
are of this
computational
inan
[1976],
type.
The
efficiency
m
with
a potential
for sufficient
accuracy
A Model In 40,000 tied
Reynolds
< R
many
number
< 4,000,000,
to a particular
In fact,
= F_
the
airfoil
different
they
range
generality.
for Airfoil over
assume
geometry types
and
Design
which many
laminar different
or to particular
of aircraft
with
separation sizes
and
conditions
different
mission
bubbles
form,
thicknesses,
each
on different requirements
airfoils. and
10
design
constraints
goals
conflict
bubbles. use
fly in this range.
with
For instance,
of natural
or 60%
airfoils verse
pressure
able
to recover
on forcing
seem
angles
however,
and,
only
if the
of pressure One bubble
therefore, effects
such
in drag, of the
of accounting used
in the
that
the the
effect
pressure
at different
Reynolds
for laminar design
and
large
the
lam-
such
that
the
advan-
of the
bubble
at first
ct's
are
starts
flight
changes
near
the
is offset,
regime.
Since
in bubble
struc-
problem
calculated
sight
obtained
distribution
of this
can be resolved
under
different
types
numbers.
separation
analysis
Reynolds
recovery
gradient
engineering
be accurately
This a "tran-
As the
higher
of pressure
can effect
is more
gradient,
would
In fact,
ad-
separating.
disadvantages
characteristic
"fine-tuning" can
the
the
layer
is between
pressure
type
encounters
of the
for thermalling
adverse
distribution
bubble
and
of such
to destabilize
therefore,
bubble.
of this
numbers
this
beginning
leads
performance
region.
of
to 50
necessarily
pressure
insufficient
ct required
in avoiding
Reynolds
distributions
analog
higher
speeds
distribution
without
gradient
trade-off,
flow at cruise
destabilizing lower
The
extensive
boundary
adverse
the
necessitates
layer
pressure
becomes
effects
high
turbulent
pressure
before
forms.
The
in the
way
transition
of attack
by the
ture
ramp
designer
The
changes
transition
laminar
the
edge.
favorable
bubble
energy
of moderately
of the
into
flight.
to help
leading
small
the
high-drag
at higher
end
layer
in thermalling
higher
design
detrimental
airfoils
boundary
trailing-edge
of a region
at the
of natural
the
flow The
the
airplane
pressure
laminar
aft-portion.
before
the
favorable
on
their
near-freestream
decreases,
a thick, tages
since
by means
boundary
over
particular
of sailplanes
The
employed
that
minimize
requirement
transition
gradient
ramp,"
number
generally
happens
would
technology.
recovery
the
that
low-drag
flow
chord
hinges
sition
the
pressure
is achieved
conditions
laminar
of the
to a steep
inar
flow
It often
program
bubbles
in airfoil
of Eppler
and
design Somers
is the [1980].
11
The
design
allows
method
great
of this
freedom
program
and
uses
flexibility
distribution
teristics
are calculated
by an integral
velocity
distribution.
Close
used
in the
distribution gram,
to achieve
the
designer
unacceptably tion
designing
airfoils
to have
predictions
laminar
separation
ing
computational
can
be used
the
drag
difficult
of modelling empirical
the
more
While cases inside
bubbles
the be the
the
recent
than
boundary-layer taken
into
bubble,
Reynolds
been
account all other
a local global
approach
has
fully
short
bubbles
approaches
of the
development in order characteristics
early
approach
rests
such This
to predict
accurately bubble
in
do
of
increascriterion
not
increase
increasingly
1984].
effectively,
a new
It combines years
with
on the
hypothesis
of laminar
useful
this
it becomes
as a laminar
upstream
of the
that
[Mueller,
thesis.
transi-
significantly
R = 500,000
decreases
might
for the presence
without
above
very
pro-
be advantageous
account
in this
information.
In this
proven
it would
developed
phenomenon
behavior.
velocity
that
more
This
inviscid
assuming
airfoils
methods.
is ac-
is predicted
increases
number
inviscid
which
number drag
charac-
bubbles
more
with
airfoil
of separation
while
airfoils
of the
development
separation
can be done
In fact,
semi-empirical
to model
which
that
by the
of the
applications,
this
detrimental
interactive
this
number
provided
has
which
Although
low Reynolds
bubble
rather
that
driven
modification and
method
aerodynamic
method
presence
properties
as the
to design
and
local
over
to design
airfoil,
be possible
through
drag
The
boundary-layer
in the
the
for low Reynolds
effectively
the
the
about
of section
of the
mapping
characteristics
performance.
transition
separation.
to eIiminate
In order
desired
the
the
boundary-layer
to aid
requirements.
of the
desired
monitoring
is warned
at laminar
the
process the
increase
occurs
should
design
conformal
in specifying
pressure
tively
to achieve
an inverse
the the
accuracy that
separation
bubble
been
separation
flowfield
speed
hypothesis
has
the
method
of it
confirmed. must
transition have
of
in some location
been
found
12
to depend
on a few local
chapters.
Furthermore,
a uniformly model unable
been
peak
and
caused
caused
the
by bubbles
In Chapter the
bubble
bubble
are
are
are compared model given
formulation.
the
data. results empirical
range
to the
3, the
large
calculation 4, possible
in the
suction
the
increase
to predict distribution
which
is of most
of the
of the
bubble
is described
In Chapter are
are
7, the range
summarized,
data
base
here.
and
necessary
modelling
In Chapter
part
and
specific
5,
Several
of the
of validity
to
of the
in detail.
analyzed
are
control
part
to the
complicated
airfoils
that
that
laminar
approaches
it is
interest.
parameters
employed
6, several
This
reduction
method
most
developed. Although
the
this
iteration,
program.
pressure
independent
following
a local
been
it is able
inviscid
operational
to model
In Chapter
important
airfoil
has
Somers
as the
in the
through
model
bubbles,
with part
interaction
and
such
In Chapter
turbulent
to experimental
enlarging
Eppler
leading
together
necessary
proposed.
is assessed, toward
in the
In Chapter
discussed of the
the
of the
in detail.
functions
flowfield
alteration
will be discussed
bubble
interactions
2, the reasoning
calculation
empirical
into
occurring
is described
for the
by leading-edge
is retraced.
of transition the
local
that
separation
for strong
sometimes
in drag
laminar
incorporated
to account
parameters
by accounting
accurate
has
scaling
airfoil
the
results
of the present suggestions
confirm
the
are
present
13 Chapter 2 LOCAL PARAMETERS
In this chapter, the approachfollowed in developingthe present laminar separation bubble model is justified. It is then shown how the shortcomings of previous similar
models
can be remedied
by letting
the bubble
flowfield
depend
on three
local
parameters.
Modelling
cess
The
development
that
is usually
variables acterize
need the
correct
iterative
conditions
tions,
which
be integrated varying
between
and
embodies
and
as for thousands Motivated
by
grees
of approximation
duced,
the modelling
about must
what
does
be supported
instance,
pressure
mass
to obtain
have
been
challenge
either
case
(2) by spatial
momentum
for the
fluid
the
of the
laminar
char-
time,
Navier-Stokes This
velocity
separation
the
flow,
variables,
conservation.
dependence
best
(3) determining
of incompressible
relationship,
dependent
variables
and
pro-
and bubble
(1) and
equa-
model
can
pressure
on
problem
as
flowfields.
computational
viscous/inviscid
the
flow conditions
what
depends,
field,
and
a three-step which
independent
phenomenon In the
entails
(1) identifying
which
(3) by a differential
of other
and
phenomenon
sequential:
two.
and
pointwise
numerically
geometry
the
the
velocity
conditions,
than
(2) identifying
on which
by the
boundary
of a physical
rather
to be modelled,
relationships
is comprised
well
of a model
Philosophy
does
eKiciency
requirements,
developed.
As soon
changes not
by analytical interaction
in nature
need
since
methods as approximations assumptions
to be approximated arguments
methods
of varying
and
have these
or by experimental rely
on the
are
boundary-layer
deintro-
to be made assumptions
evidence.
For
assump-
14 tions and on the extensiveanalytical and experimental evidencethat There
are
ber
and
two important types
methods (2)
of flowfields
is severely
by
the
shifted
from
arrived
at by allowing
the
and
them
valuable
then
The
Navier-Stokes
will be even
and
closer
imation
process
of the
phenomenon,
if such
a solution
more
limited
to the
integrated
solutions.
explicit
necessarily is not The are
in applicability.
yond
the
viscods/inviscid
case,
this
approach
has
interaction proven
at
Laplace's
the
their
is roles
relationship equation
is limited
flows
above
solution
the
information
in this
or
to the
way
knowledge
thesis,
physical
adds
that
the
it is
and may
to be a successful
limit
to this
have
brought
become
model That
is brought
between
the by
too
to a semi-empirical compromise
approx-
mathematical
of approximation, insight
to
pointwise
of its generality.
approximation
approach
it will have
differential
original
present
is necessary.
and
the
expense level
as in the
phenomenon
the original
to the
that the
stringent,
In essence,
of speed
by a model
In this
of either
more
from
at a particular
advantages
off-set
with
in applicability
solution
obtained
possible
iteration,
and
even
of the
removed
further.
however,
are
are further
an
final
are
equations.
that
is simply
The
viscous
equations;
conditions
to interchange
solution
over
original
flows
number
solution
approximation
balance
restrictive
the
a more model
the
Reynolds
requirements
drastic
integration
Although high
by the
at each
num-
boundary-layer
independent
to flow variables.
boundary-layer
numerical
about
the
by the
them.
(1) the
interactive
be done
problem,
processes
the
insight
on relationships
solution,
and
with
could
variables
computational
resulting
is simplified
outer
of approximation:
analyzed
what
spatial
approximating
pointwi_e,
If the casc,
by
be
or outer
equations.
integral
governed,
rely
given
surface,
can
independent
boundary-layer
airfoil
and the
of this type
from
inviscid
geometry
as dependent
that
restricted
integrating
between
consequences
supports
is,
model
an explicit
simplistic
and
one
be-
method. speed,
step
In this gener-
15
ality, the
and
understanding.
governing
integral
complemented bubble
eral.
This
adding
the
formulation only
prediction
course
of the
predict
the
effects
inviscid
flow began
[1967],
modified
Mueller
[1989],
in terms
of integral
suited
to the
method
with
into
that
integral
here,
and
of the
Somers
ds
with in the
of Thwaites obtained local the
appropriate
directly
non-similar
program.
interacts
weakly
empirical
model
[1980]
Because
bubble
models
method
differential
a method
and
they
employed
equations,
the
the
of Horton and
formulated
as these by
to
with
Schmidt are
such
able
are well
Eppler.
This
momentum
and
(H. + 2)U dV ds
2
-
Contrary
gradient
the
boundary-layer
d--_
relations
for
methods
governing
parameter.
CD -- 3-_
to simpler,
in two-equation from
(2.1)
53 dU
closure
Appendix. [1949],
pressure
which classical
analysis
d53
given
and
equations,
dh _ ds
together
models
necessary.
to develop
of Roberts
properties,
governing
absolutely
efforts
bubble
suggestions
boundary-layer
two coupled
it was
in gen-
Results
incorporation
Eppler
with
are
features
simpler
starting
separation
boundary-layer
integral
employs
energy
the
essential
the
by
reported
to the
the
and
at
establishing
according
capture
bubble
of its effects
after
the
that
in the
"guesses,"
and
arrived
of a laminar
on empirical
of its characteristics
was
research
reliance
are enforced
relationships
Early In the
the
equations
a set of simple
allowing
complexity
to minimize
boundary-layer
with
of the
In order
equations This
allows
developments
(2.2)
c$,
CD,
and
one-equation the
shape
factor
and
is therefore
such
methods
characteristic
H12
[Eppler,
1963],
such
as that
methods (H32,
in this
independent to analyze
case)
is
of the accurately
of aerodynamic
flows
16 provided
that
reasonably
empirical
by
along
assumed
family
of velocity
profiles
approximates
the
actual
models
bubble.
of rather pressure
above
method
Instead,
coarse
they
between
by the
do not
to calculate obtain
approximations
plateau
brought
noted
a two-equation
the
the
near-stagnant
Thus,
region
the
shear
bubble
assuming
and
leads
accuracy
of the
of 52 along
transition
in this
of the
development
equations.
and
fluid
advantage
the
growth
of these
separation
take
negligible
the
ments
momentum
[Schmidt
Mueller
and
[1989]
integral
equation.
Mueller
1986;
suggest
using,
by means a constant-
skin
friction
to
the
tum
thickness
Russell this
similarity
From
length
in Chapter the
transition
is approximated
Combining
these
_
(62)s o
for the
predicted
increases the
be discussed
ble
growth
[1978],
equation,
solution
with of the
by
laminar this
laminar
4, must
on
low Reynolds
1986;
Brendel,
by Horton
free
shear
equation, Reynolds part
the
leads
growth
of the
the
momentum
Iayer.
[1967],
The
was
number. bubble,
value
proposed In order
given
turbulent
and
to Truckenbrodt's
it appears
from
and
of momenearlier
by
to evaluate
by correlations
part
energy shape
experimental
=0
v
Schmidt
(2.4)
which
of 82 in the
ds Ira w
1988],
measure-
(1"1969)2(gl/c)
" 62 dU 62dH32 G - (H12 - I)H 2F-a-7 + As discussed
number
to
be known.
point,
equations
_/1 +
decreasing
by simplifying two
O'Meara,
(2.3)
instead,
(62) c from
Based
aflayer
(52)7- = (52)s from
flow
well.
The forded
the
of the
bub-
integral
equations.
parameter
equation,
e fH32
(2.5)
2
data
that
(2.6)
17
Using
this
tachment,
result
Horton
along
was
with
able
that
to reduce
Eq.
U--d-s-s _ The
assumption
turbulent
of a universal
part
Up
and
can
integrate
of the
H32
and the
leads
1)
recovery
profile value
from
equation
friction
at
a point
of reat-
n
velocity
to a constant
integral
skin
to
Ha2(H12-
pressure
energy
(2.5)
reattachment
bubble
a linear
of vanishing
and
a constant
for A_.
Assuming
transition
to obtain
the
CD in the constant
to reattaehment, momentum
one
thickness
at
reattachment,
(5:)n
w
Then,
=(52)z
eliminating
+C2
\U--_n)
(52)_
between
(4---'_32)(1+
Eqs.
(2.7)
£2 =
In order from
the
of Uz'.
segment
joining
the
this
happens,
This account tribution
(52)_
is started type for the in the
the
neither
Has
local and
relies
influence nor
the Eq.
value
obtains
on
of the CD
are
is decreased
and
(Ha2)=
therefore
cannot
empirical
upstream constant
whether
distribution.
turbulent
boundary-layer
reasons.
It does
not
interaction
on the
pressure
accurately
calculate
the
boundary-layer turbulent
criteria
conditions.
which
development. part
or
When
for initial
transition
in the
increments
it is checked
velocity the
in small
= 1.51
for several
viscous/inviscid
local
and
inviscid
(2.8)
and
is inadequate
part It
from this
Us
g2 is calculated
intersects
is calculated
laminar
able
step,
7" to _
of the
one
j
(2.9)
numerically,
At each
effects
region.
to sense
result
of formulation
(2.8)
\_-_-)
Gj?_
at are using
of 52 in this
actuality,
this
value
not
method
w
to implement
and
U_--_-) 1+
of the
properly disgrowth are
un-
Also, bubble.
in
N
18
The
turbulent
Finally, the
the
pressure
corresponding In
the
recovery
In the
at the
at
van
laminar
part
appreciably
point
approximating
functions
use
deviate
from
a straight
can be significantly
below
line.
or above
value.
better
Ingen's
can
reattachment
inviscid
an attempt
bubble,
distribution
were
is made U
-
Us
the
pressure
implemented
of the
empirical
distribution
[Ingen
and
induced
Boermans,
by 1986].
function
.978 + .022exp(-4.545_
- 2.5_ 2)
(2.10)
where _-ss
(2.11)
¢ = (R 2)s(6 )s which the
allows
for a slight
turbulent
assumed Fig.
part,
that
2-1
dashed
Using
the
to capture
gradient length bers
while for
but
to small
not
ble
of predicting
dictions.
empirical
in pressure
variations
beginning
based Fig.
the
in the
of the
achieve
only
model
prediction
accurate
chord
Reynolds
number.
models
responds
to variations
gradient.
The
governing
of the
the
drag
parameters, recovery.
bubble
to within
at separation
aerodynamic
by Eq.
the
parts
thirty
cannot
characteristics
the
although percent,
of the
of
of the
the
bubble
it was
that
in turn,
provide
(2.9).
adverse
in chord
Thus,
The
sensitivity
It is found
for instance,
pressure
recoveries.
increasing
prediction,
it is
distribution.
predictions,
with
For
again
inviseid
various
polar
bubble
transition. and
Stratford
noted, to
drag
the
as given
of the
on conditions
2-2 shows
and
vanishing
turbulent
features
drag
with
points
bubble
and
is employed
Horton
reattachment
and
to
distribution
the
separation
at a constant
such
the
models
of possible
In order
separation
at the intersection
between
development
explored.
necessary
occurs
empirical
boundary-layer was
locus
between
pressure
a comparison
is the
recovery
a Stratford
reattachment
shows
line
pressure
found
pressure
the
transition
Reynolds
num-
is very
sensitive
pressure
level
generally empirical
acceptable Eppler
at
capabubble
drag E387
preairfoil
]9
m
0 o F-q
_o
0 0
0 0
0
!
0
c_ 1
0
_,
II Q)
_n 0 o
tY)
0 o o o o
(0 v
Id n,
21
at R = 300,000 the
original
[1988]
obtained
Eppler
taken
with
and
in the
this
Somers
early
version
prediction
Low Turbulence
of the
and
Pressure
the
bubble
model
measurements
Tunnel
at the
compared
of McGhee
NASA-Langley
to et al.
Research
Center. Before tions,
beginning
two issues
due
to the
does
not
equal
to drive
the
of the
predicted
and
discrepancy
can
measured be
While
appropriate
are not
really
necessary
concentrated. the
same
experimental dictions
and
obtained
polar.
It may
mid-chord
at lower since
than
locally.
the
if the
same
employing
inviscid
on highly
the
the
near-stall
where
poses
validity
aft-loaded
inviscid
the
to a discrepancy
between
is prescribed.
This
interaction
algo-
methods
design
in angle
no obstacles
of the
airfoils
and
it will modify distribu-
are
distribution
it, leading
pressure
conditions,
difference
pressure
is present,
Firstly,
distribution
falls inside
of attack
characteristics
a, the
but
leads
angle
approxima-
pressure
a viscous/inviscid
lift coeffcients
same
that
Using
necessarily
lift coefficients
be expected
bubbles
bubble
above
be addressed.
experimental
interacting
aerodynamic
at the
inviscid with
by
the
should
of attack
therefore, results
model
to the
angle
for analyzing
In fact, ct rather
only
eliminated
rithm.
layer,
If a weakly
model,
alternatives
bubble
one at the same
distribution
the bubble
the
boundary
coeffcient.
pressure
of possible
affecting
the inviscid
lift
viscous
tion
indirectly
presence
to a smaller the
a discussion
effort
of this is most
often
usually
compared
at
of attack
between
the
drag
pre-
to comparing to the
present
type
experimental
model
will
for leading-edge
drag
decrease bubbles
for near
bursting.
and tween high
Secondly,
the
employed
in the
the
integral
Reynolds
turbulent program
variables
number
flows
boundary-layer is based
analysis on empirical
[Eppler,
1963].
without
bubbles,
While such
method
developed
equilibrium quite
relationships
appropriate
a method
by Eppler
cannot
be-
for analyzing correctly
ac-
22 w
count
for
the
reattachment, makes
use
Drela
relaxing,
nonequilibrium
especially
at lower
of the
[1986].
ods,
in Figs.
are
analyzed
nal
and
predicted
is consistently
It is stated
parameters,
may
be
the
to capture
it has the
flowfield
been local
along
exception
flowfield.
have
the
correct
lowest-order
The
failure
or success
response of previous
transition
NLF(1)-1015
airfoils
using
the
separation
for the The
must found
origipoint
low Reynolds
drag
coefficient,
be able
of the
of the
approaches
early but
other
of the bubble to these to predict
also
of the
the
bubble.
models inability In spite
of this aspects
bubble mean
inputs
accurate
present
three
of
model. flowfield. of the
characteristics: inviscid
pressure
In addition,
must
bubble
of
transition
to their
at transition.
the
by means
an
in the
to characterize all
hypothesis
empirical
prediction
at separation,
bubble
solely
without
abandoned
detailed
the
Although
help
two parts
to control
thickness
confirmed
of transition
a more
Bubble
process.
failure
not been
thickness
the
bubble
of much
in the
has
to achieve
the
meth-
turbulent
has
interactive
modelling
conditions
and
by
boundary-layer
number.
model
the
processes
been
therefore,
developed
point
Controlling
it is not Thus,
poor
momentum the bubble,
of the
effect,
parameters
that
boundary-layer
gradient
weakly
extended
separation
Parameters
the
on local
NASA
of
10 to 15 counts.
to model
physical
the
by Eppler
present
only to their
reliance
SpecificMly,
the
not
the principal
their
bubble
bubble
two
predicted
the
the
these
Reynolds
model,
method
The
higher
downstream
present
methods.
1 that
important
of the
can be traced
Rather,
with
most
by
Independent
be possible
locM
estimation
higher
in Chapter
it should
of that
to it at the
The
and
laminar
boundary-layer
is equal
Local
E387
at the
is downstream
layer
boundary-layer between
Eppler
transition
turbulent
The
turbulent
the
boundary
numbers.
a comparison 2-4
assuming
case but
however,
this,
and
by Drela
number
that
provide
2-3
Drela's
Reynolds
nonequilibrium
To
turbulent
be identified.
flowfield
and
its
23 w
t-0
C: "--'
0 X
C _,_
,,,, a;
>,-, 0
>-. C_
4_
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C.O 0 "0 0
I I C c:
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25 w
effects
on airfoil
success
to account
The
first
location They the
performance properly
recognized
boundary
layer
at separation, one
would
has
universally
flowfield
is.
becomes
longer
a good
flow
could
In
need
as the
it bursts
correlating
with
the
at the
parameter
for the
that
long
from
stability
of the
boundary
Reynolds
layer an
number
a particular the
plausible,
separating
in
whereas
is decreased,
of the
bubble.
present
turbulence
It seemed
or
transition
bubble,
The
number
value.
size
stable
bubble. far
the
of instability
A very
failure
conditions.
the
by a fairly
Reynolds critical
flow
degree
surface.
of how
chord
local
to their
recognized
a shorter
as an indicator
addition, until
to estimate the
related
in determining
to be followed
associated
used
and
problem role
from
is likely be
independent
important the
to be directly
bubble
as it separates
in fact,
unstable
that
of the
a fundamentally
therefore
be shown
for these
investigators
plays
been
can
bubble
therefore,
boundary-layer
m
velocity and
be arrived and
at by forming
momentum
thickness
a Reynolds
number
at laminar
with
separation
the
value
of inviscid
as characteristic
velocity
length,
(R6.)s-
us(6 )s
(2.12)
/./
This
parameter
tion
of the
been
used
stability
of the
extensively
at laminar than
is certainly
flow
but,
unfortunately,
at laminar
in empirical
separation
a measure
useful
separation.
correlations
nondimensionalized
of the
stability
of the
only
Another
is the
with
value
respect
separating
provides
a coarse
indica-
parameter
that
of momentum
thickness
to the
airfoil
chord.
layer,
this
parameter
shear
has
Rather only
m
w
contains
information
ary layer
upon
to the stream
effect
on how much
reaching of the
the
pressure
boundary-layer
laminar
momentum separation
distribution
development.
has point.
on transition,
already These
namely
As will be discussed
===
is well
modelled
by the
e" method
of linear
stability
been
theory.
lost by the
criteria
lack
the
effect
in Chapter
bound-
sensitivity of the
4, this
up-
effect
26 Another
effect
interaction. since and
Since
the
of separated
that
another
to calculate
ber
flow
of the
laminar
separation
lence
cannot
to the
grid
within an
reasonable
adverse
lower
presence
effect
has
of the
been
conditions,
viscous/inviscid
of separated
by the severity
that
found
flow
pressure
and
gradient
to scale
Gaster's
unsteady
study
parameter laminar
a duct. between
well
pressure
with
gradient
Navier-Stokes
used
to keep
the
A suction
port
on
upper
for the
local
laminar
and
transition
layer
are num-
tunnel
the
to turbu-
can
computational
boundary
in yet
Reynolds
of the
wall
1989]
equations
unsteadiness
time-step
gradient
the
entrance
(computational
the
et al.,
to be important
Although the
[Pauley
of R = 50,000-300,000,
computed
limits.
a recent
Gaster's
is on the order
and
pressure
the
distance
accurately
resolution
point
flow through
on the
point
be
flow
shows
study, the
based
these
bubble
In this
discretized
this
at this
separation
respect.
that
strong
(1.4).
be mentioned
laminar
of the
flow is affected
surprising
by Eq.
for is the local
is a result
incorporates
given
It should
to be accounted
interaction
it is not
a parameter parameter,
needs
this
amount
by (52)s,
of the
that
occur)
due
requirements
of the
duct
provides
developing
along
the
wall. The
with
value
the
of the
velocity
parameter
gradient
P,,_a_ = .24,
representing
the
a modification maximum
to Gaster's
rather
than
parameter
the
average
in-
=--
viscid
gradient
boundary This long
between
boundary (steady,
by Gaster. unsteady process
between steady
correlates
and
short
of this
well
reattachment,
unsteady
the
short
the
large-scale
of the
bursting
line
(unsteady,
between
it is proposed by
is found
reattachment
to Gaster's
correlation
bubbles,
be governed
and
and
P_,,a_ < .24) from Because
may
separation
the
P,_a_ steady
such
to correspond
to the
laminar
layer.
that
shear
it separates
> .24) bubbles and
by Pauley
et al.
laminar
pressure
long that
bubbles the
field
the
measured and
the
reattachment rather
than
by
27
turbulent
transfer
certainty,
the
fact,
onset
as Pm,_
ration
increases.
laminar
shear
ably
occurs shear
the
Thus,
pressure
layer
This
by Russell
[1978].
When
of the
below
the
large
vortex
flow
that
is, the
and
is shed
growing
causing
again
Before
and
the
shear
above
bubbles,
too,
were
thought
that
the
transition
it should
not
necessarily
not
effect
be given
depend
mainly
therefore, so well may
with
why with
have Given
the
more
arrived
large
the
prob-
region
below
is discussed
without
suction
bounds;
becomes
small
the
As the
favorable
pressure
This
used
laminar
for a complete of approximating
it was
and
until, unstable
bubble
than
short-time
that and
then
understanding
the
final
model,
downstream
of transitional
if there
the
present
simulation
feed
that
that
starts
back
bubbles. of bubble
the viscous/inviscid
It was
also
upstream
and
is some
unsteadiness
turbulent
transfer
reattachment word
of than
on transition
it appears
conditions.
reported
transitional
mean.
in determining
Although
in the
thought
would
now
factor
is present.
measurements
scale
in nature
significant
method
at,
reattachment
be periodic
the
a means
flow.
bubbles
growing
in size.
It seems
rather
to wait
starts
at scpa-
reattachment
exceeds
flow at constant
near
on upstream
Gaster's
of reverse
separation
In
to recover
of a low-pressure
gradient
flow.
loss
is necessary
transfer,
to laminar
was
location.
unsteady
momentum
amount
formation
bubble
of laminar
with
repeats.
in the
if turbulence
sizable
or disproved
case
the
transfer
to collapse
unsteadiness
is a much
the
bubble
unsteady
the
Coanda's
layer,
cycle
and/or
inviscid
proved
in the
momentum
or the
adverse
description
influence
momentum
of such
of recirculating
the the
the
as it relates
the
be
be justified
gradient
effect,
effect
cannot
momentum
capable
to Coanda's
layer.
inviscid
to accelerate
is not
this
can
a greater
and/or
due
While
of unsteadiness
increases,
inviscid
the
of momentum.
at this It is still
by Pauley
et al.
Resolution
canpoint
not
clear,
correlates
of this
issue
bursting. interaction
to
in the laminar
28
part
of the bubble
can be estimated. bridges
the for
the
weak
function part
the
This
laminar
length
turbulent
and
latter.
transition
geometrical
and
the As
of Reynolds of the
bubble
location, characteristic
turbulent
the
the
number follows
of the
parts,
spreading and
providing
angle pressure
directly
height
and
of the
of the bubble the
is the correct
turbulent
gradient, the
bubble
model
in fact,
at transition keystone
that
characteristic shear
the
is thereby
layer
length closed.
is a of the
29
Chapter THE
The
Eppler
separation.
and
LAMINAR
Somers
It is based
PART
program
on
the
3
uses
value
OF THE
a very
of the
BUBBLE
reliable
energy
criterion
to detect
to momentum
laminar
thickness
shape
factor, (H3_)s---This
value
is approached
development and
of the
energy
from
separated
integral
above.
laminar
equations,
Upon shear
together
Instead
of implementing
this
usually
done,
the
development
laminar
part
of the
bubble
1.515095
of the and
function
bubble given
laminar
used
its calculation
by Eq.
(2.10).
separation,
the
of that
This
approaching
bubble
region
NLF(1)-1015
airfoil
in the
NASA-Langley
that,
as the
tends
(2.10)
pressure
to fall below
becomes is therefore
flatter,
checked
gradient
along
van
closer
postulated
was
Ingen's to Horton's that
Eq.
allows
available
the
in the
Ingen
a slight
several
Using
wind-tunnel
bubble while,
can
different decreases, as the
as it is in the
be improved
laminar
Boermans
above
[1986]
detailed tests
conditions. the
part
recovery
after
pressure
of the
Pressure
pressure
and
and
pressure
value.
from
approximation (2.10)
mode
mode.
Low-Turbulence
for
curve
by van
a limiting
in the
of Eq.
direct
below.
distributions
distribution
developed
distributions
accuracy
inverse
of pressure
pressure
distribution
quickly
momentum
to be discussed
in the
in the
the
the
Distribution
to approximate
is a generalization
family
separation,
using
relations
method
of a general allows
of laminar
is calculated
closure
boundary-layer
Pressure The
detection layer
with
(3.1)
NASA
Tunnel, It was
the found
pressure
distribution
gradient
steepens,
van
by relaxing
it
Ingen's
curve.
It
the
amount
of
w
30
pressure
recovery
between
separation
and
U____= (1 - DU) Us The
steeper
the
This
behavior
pressure
by use the
along
with
an inviscid
displacement
While
the
of Eq.
agreement (3.2),
pressure
+ DU exp(-4.545_
gradient
is consistent
ever-thickening
surface with
the
over
the
- 2.5_ 2)
bubble,
the
velocity
experimental
leading-edge
smaller
the
adverse
apparent bubbles.
value
of DU.
calculated
over
an
gradient.
distributions
became
(3.2)
distribution
in a steepening
an inconsistency
distribution
transition,
was
when Given
much
improved
attempting the
very
to predict
large
gradients
r
along
these
quite
flat,
of the and
bubbles, in contrast
inviscid
the boundary
layer
there
can only
a very
inviscid
hold
distribution
Although at
value
of (62)s
separation
determines layer,
that
imposed average are
is, the inviscid
pressure
included
that
small
amount
the
trend
the
by the
short
bubbles
of fluid
and
for mid-chord
of the
(1-
can
separation
of pressure
bubble.
recovered displacement
Near
6%c)
the
usually
therefore
were
perturbation
observed
amount
part
between
perturbation
entrained
is so thin
is a strong and
more
such
of fluid
a small
recovery
the
of the
only
laminar
only
leading observed
modify
the
slightly.
there
layer
that
magnitude
amount
show
with
it is realized
in the
pressure
contradiction
once
edge,
which
a significant
to the
is, to the
distributions
measurements
apparent
proportional that
pressure
with
be resolved
is inversely surface,
to the
This
can
predicted
distribution
transition.
bubbles
the
the
correlation amount
precisely
between
of fluid
reflects
the
the
caught
input
thickness
in the
to the
separated
the
greater
the
greater
the
amount
of separated
flow necessary
DU
therefore
gradient
in Gaster's
The along pressure
variable the
bubble gradient
already
should as well
as on
parameter,
lost
(62)s. Eq.
boundary region,
momentum
amount:
gradient.
momentum
of the
the
balance by
the
that
boundary
to counteract depend Both (1.4)
the
both
on
these
effects
which,
for
the
this
31 reason,
is thought
dimensionless
to be
inviscid
a better
velocity
independent
gradient
parameter
along
the
than
simply
the
average
bubble,
A(U/U_,)
(3.3)
/,(s/c) In dimensionless
variables
P becomes,
p =/_
(5_______ss
From sented
experimental
as a function
relationship, of DU
of the
shown
in Fig.
and
P directly
1015
[NASA
LaRC
1988].
The
solid
pressure
from LTPT,
distributions,
Gaster
pressure
3-1, was the
it is found gradient
developed
1987]
line
is a quadratic
DU
= _ 0.0610 [ 0.0152
the
Eppler
least-squares
DU
parameter,
pressure
and
that
P.
by extracting
experimental
June
(3.4)
x(s/c)
o
is well This
repre-
functional
corresponding
values
distributions
of the
E387
[McGhee
et al.,
included
in the
fit that
airfoils has
been
NLF(1)-
model,
The
value
the
variation
of DU
= 0.022
in DU
As pointed
out
shown by van
ensure
continuity
in the
method
is used
to determine
of Eq. one
(2.10)
in Fig. Ingen
is evaluated
Ingen
and
2
-P -P
< .3 > .3
Boermans
falls
(3.5) in the
middle
of
derived
to
3-1. [1989],
gradient
the
+ 0.5072P
laminar
at laminar
the
factor
when
4.545
in Eq.
Thwaites's
separation separation
laminar
point. and
the
(3.2)
In fact, variables
was
boundary-layer if the are
derivative rearranged,
obtains
which (3.1)), for
by van
velocity
[ r
used
+ 0.3048P
is Thwaites's however,
differing
-2_v dU] _s s =-(4.545)(.022)=-.10
laminar corresponds
upstream
separation to
slightly
developments.
criterion. different Furthermore,
(3.6)
Eppler's values
separation of Thwaites's
in the
present
criterion
(Eq.
parameter formulation
a
,"]2
z
0 r_
@ 0
40
c)
I:L
(::)
o []
_°
0
g g _
II
o.
I1_
o
0
II
°
_.x ILl 0
_:> _
Z D
5-_
0
:3
o_
._
33
variable
DU
a continuous with
Eppler's
is used
in place
velocity
gradient
laminar
of 0.022.
Therefore,
at separation
boundary-layer
the
when
method
constant
Eq.
necessary
(3.2)
is used
to ensure
in conjunction
is
act C = -DU When
C is substituted
for 4.545
U Us where the
the
prime
exponent
of the
in Eq.
(3.2),
s the
_l_DU{l_exp[2uU_S(s_ss)]}
denotes
the
derivative
original
(3.7)
-
has
is
(3.8)
Us
with
expression
result
respect been
to s and
the
second
term
in
neglected.
Closure In order mode
to integrate
as driven
H12,
c I, and
Co
Falkner-Skan, Hartree,
by
must
or
profiles
of separation. seem
was
measurements [1966] a base.
[Schlichting,
1979]
axe used
from plitude
the
measurements the
therefore and
Mueller the
centerline of Coles's
two-parameter
in Fig.
3-2,
profiles. of the
wake
wake
function.
above,
natural
equations closure
choice the
to develop
may
is to use
the
the
correlations
be
for
reversed
Falkner-Skan,
Mueller
not
direct
correlations
attached
and
in the
the
or
upstream
[1990], best
however,
choice.
This
between
their
detail. obtained
forming
(h/b)
given
since
profile
the
integral
Fitzgerald
have
layer
energy
profiles
in some [1990]
shear
of the
by
Stewartson
examined
a turbulent
characteristics
most
profiles
that
As shown
The
[1954],
and
for
distribution
be provided.
Recent
Fitzgerald
pressure
and
Stewartson
to indicate
matter
the
the momentum
two
good
family a free
originally
ratio
developed
stagnation
parameters
is the
agreement
are of the
point
linked distance
to the
width
of the
shear
layer
Since
there
is slip
along
the
by Green
downstream
to the
geometrical
of the and
of
shear
G is the
centerline
layer am-
of such
a
34
0 r._ O ,--4 |
>-)
,I
0 c_ >-
b_
> )-(
c_ c_
I
f,_
,i,,m
°_ =
_
35
recirculating
base
By
the
applying
for the
flow,
these
definitions
dissipation
profiles
cannot
for the
integral
thicknesses
of the
following
relationships
are
coefficient,
the
H12 =
be used
to develop
a correlation boundary
for cl. layer
and
obtained:
1 + 2_
(3.9)
3a) - 2 (1 - 2a) H32- --
(2-9
7G + __G _) + 4-}(1 - 3G + 2G 2) (1 - 3_G)2-}(1 - 2G)
3
_2_3
In order profiles, values can
it is necessary used
of the and
then
repeated
for values
in order
to determine is shown
are
compared
solid
lines
abscissa
quite
different While
very
(h/b)
(h/b)
developed
separation
further
downstream.
thereby crucial.
where
these [1986]
of G and and
corresponds
seems
correlations
transition,
to moving
Stewartson
quite making It appears
from
downstream
sensitive the from
to
the
are and
changes
then
Mueller The
correlations profiles.
H12 and
inside
inside
parameters.
to greater
near
determination
calculations
Stewartson
moving
evaluated
station
two-parameter
As both
The
one bubble
by Fitzgerald
the
correlations
bubble.
are
downstream same
Stewartson
inside
to these
new
(h/b).
the
variables
used
of the
the
inside
measured
The
to those
by Drela
variations
vary
same
another.
similar
3-4
from
boundary-layer to the
one
sensitivity
to the
less
two parameters
three
(3.11)
obtained
to fit the profiles
against
similar
is not,
to those
corresponding
3-3 and
fitted
plots
H12(Ha2)
CD(Ha2,Ra2) on G and
the
the
on these are
The
of G and
between
both
point.
in Figs.
utilize
monotonically
Mueller
plotted
to those
how the
and
parameters
h (1 - 2G)]
relationships
to know
as a starting
bubble
result
these
by Fitzgerald
serve
at values the
to compare
(3.10)
The
H32 increase values
the
bubble.
separation
but
of the Thus, can
in
the
parameters,
of its
exact
dependence
measurements
that
the
back-flow,
be
36
o t_
0 6.
"0
_r_
t_ 0
°_
0
_0
o
_f
0 0 {13
0
o o
o_
37
0 .°°°.
r_
l i!i/i ! _.iiiii_. I ::::i ,,
(D i
,-ul
.°,°
I
7:::: o 1:111
i _;f
0 i
N
i -_i!
"1"
0 l.)
0
i:::::,,.,_ '-0
•
I
OE_
0
I
0
i :D :: I
,
I
,
I
,
I
I
0
_
0
0 e4
o
38
proportional
to G, may
be constant
ent
As shown
in Fig.
bubbles.
although stant
different value
between Ha2,
Eqs.
(3.9)
calculated
bubble
value,
(3.10)
the
easier
the
each values
follow
velocity.
and
from
appears
3-3,
in absolute
of back-flow
within
the
These
expressing
governing
equations to local
3(1 - G) -
although
of shape
factors
same
slope,
considerations
and
to correlate
bubble
the
for differ-
actually
thus justify
closure
and
different
measured,
confirming
a con-
eliminating
(h/b)
relationships
G, whose
in terms
behavior
within
each
flow conditions,
H32
(3.12)
H12 = (1 - G)(I - 2G)
Rs,
Although oped
_r2G 2 a [ 1 -- 3-_G -
--
very
encouraging,
to be implemented
of G on from not
CD
local
these well
from
flow
represented
them
profiles.
are
until
these
oped
by
by
the
issues Drela
better, have [1986]
equations
yield
in their
is unknown
contrast,
not far from
It seems
integral
By
results
model
conditions
profiles.
(4 - 5G)(1 4(1 -
these
in the
therefore, been
no
values
to keep
resolved.
using
been
implemented
the
value
of Ha2 directly,
of c/
of the
integrated
the
in the
can
with
be
obtained
profiles
the
fitted
the
factor
are
derived Green for now,
correlations Since
shape
devel-
dependence
profiles,
closure
model.
Drela's
the
parameters
Stewartson the
well
velocity
obtained
Accordingly,
have
In fact,
measure
the
(3.13)
]
sufficiently
form.
details
profiles,
(2 - 3G)Ha2 2H32
appear
present
the
corresponding
G)G)
do not
and
although
Stewartson the
of
develgoverning
correlation
is
inverted, l
= The
skin-friction
0.08
coefficient Cf
w
H32 1104008 [(H3 110 008 ]2 +
is found
.-_
/
-64.4
(3.14)
from
f
-0.067
+ 0"01977(7"4-H12)H12-1 2,
H12 < 7.4
R62 -_- = [
-0.067
+ 0.022
H12 >_ 7.4
[1
H12-6 1.4 J 2 ,
(3.15)
39
while
the
dissipation
coefficient
is given
CD
0.207
__
R62 H3---2
Removal Knowledge simple
technique
to allow inar
for some
direct
of the
pressure
effect
mode
the
of the
When
the
from
of the
separation
remain
before
starting
to grow
normally.
in the
following
sense.
aration
does
by
the
singularity
steeper
point
values
continuous much present again
steep
behaves
upstream
aration
of the
with
is therefore
its
the as the
calculated
laminar
separation
(cubic)
development
than slope
is believed
local
thereby
upstream, cannot
where
actual
laminar
again point
of H12(s)
separated
a few point
and
solving
point. chord
prescribing the
The
using
laminar
The
as,
layer
in fact,
growing
boundary
them
boundary-layer
layer
"inviscid" an
a
In the
starts
of the
sep-
gradient
of growth.
laminar
between
the
a pressure
rate
upstream
separation
when
layer
shear
of H32
skin-friction
of the
Therefore,
previous
to a much
distribution
boundary
laminar
percent by
the
the
separation
from
to this
by
affected
is equivalent the
chord
of sep-
is increasingly
causes
location.
maintain
the
but
of separation
it is felt
the
singularity,
upstream
to a prediction
observed
above,
of the
of separation.
leads
in
for a few percent
effect
and
upstream
downstream
which point
present
immediately
experimentally
value
This
and
is calculated
variables
gradient
to a
of lam-
described
a consequence
boundary-layer
pressure
is actually
and
values
led
upstream layer
function
separation
of the
shear
has separation
distribution
the
is approached.
incremented
gradient
model, is taken
the
similarly
are
gentler
just
gradient
a very
coefficient
growth
as separation
pressure
to exhibit
This
at laminar
laminar
using
at their
of separation
singularity
development
variables
reflect
downstream
Goldstein
(3.16)
- 4) 2
Sing_
on the pressure
boundary-layer
not
Separation
bubble
laminar
The
- 0.003(H12
distribution
for removing
separation.
the
m
of the
by
assumed equations
4O in the
inverse
point
mode.
growth
from
local
the
tion
and
as the
the
to the
than
the
is calculated
Using
this be
the
separation
to start
deviating
corresponding
the
velocity
assumptions
inviscid to Fig.
original,
pressure
this
new
As this
of the
separa-
behavior
to be correct.
as separation
upstream
separation
of DU
point.
is believed
down
distribution
old value
separation
distribution break
3-5, as the the
laminar
is approached,
of separation
point
is usually
is usually
too
great
is to be at a higher
and
a new
one
from
value
used
Using
and
the
current
= DUotd
new
to generate
the
tangent
value
a new
distribution
new
of the
by an empirical .
through
distribution
measurements,
-
(Us)old
velocity
at the
of momentum
angle
the
whose
tangent
Eq.
(3.8)
is continuous
Angle at the
streamline
proposed
at separation,
point.
thickness
separating
(3.17)
gradient
distribution
separation
value
relationship
-- (Us),e_ (Us)ola
for the velocity
Separation
the
layer
velocity of the
experimental
DU,,e_
with
boundary
the
upstream
boundary-layer
Referring
pressure
causes
distance
of the
a modification expected.
solution
some
matches
point
In fact,
inverse
inviscid
distribution
can
of the
results. The
u
Smooth
"viscous"
makes
by Wortmann
separation
with
the
surface
point, is given
[1974],
i
64P (3.18)
= - (-a6,)s A similar
relationship
proposed
by van
Ingen
et al.
[1980]
was
also
examined,
B
t=-y_ B is given
the
experimental
mean
value
(R6=)s of 17.5.
(3.19) The
model
did
not
perform
at all
N
well with
m
the
latter
relationship.
The
reason
is that
the
factor
64P
assumes
a wide
.,.
_. 0
o
N)
0 0
o 0
E X 0
°_
0
.e
w °_
0 o _
°_
0
0 Z w
I
,011 m
o
o
m
,-
0
D
0
_4
_
42
range
of values,
their
average
hand,
gives
from does
as low as 2 up
not
give the
to 30 for the
model
enough
range
flexibility.
of bubbles Eq.
examined,
(3.18),
and
on the
other
w
the
for /3 as given Navier-Stokes
r
correct by Eq.
simulation.
scaling (3.18) Eq.
for this have (3.18),
been
variable.
In
reported
therefore,
addition,
by Pauley has
been
very et al.
included
similar [1988] into
the
values in their model.
43 =
w
Chapter
4
TRANSITION m
w
The deal
prediction
of the
of attention
the
bubble
the
calculations
flow.
since
bubble
found
switch
must
been
criteria
well
are
inside
the
bubble
models.
In fact,
both
the
behavior
depend
strongly
first
bursting
or
claimed
be modelled very
location
in a more
it has
to work
transition
the very
as well as the
Although
the
transition
less that
accurately
in the
the
empirical bubble
since
bubbles
correlation
were
together
between
characteristics
such
first
with
a great
increment
on the
of the
due
location
laminar
to turbulent
transition
region
chapter,
method
to
where
a point-transition
In this the
drag
from
1989],
received
has
a few
employed
inside been
empirical
in the
model.
looked
for
Criteria
observed,
the
length
model.
Empirical Ever
way
[Walker,
present
discussed
gradual
has
many
distance
as conditions
from
researchers
separation
at separation.
have
to transition Since
it was
and
observed
an
local that
_l
_J
the
length
transition w
bubble
criterion,
number and
of the
based
is inversely
proposed
on
the
proportional
by Von Doenhoff
velocity
at
separation
to the [1938], and
the
Reynolds
number,
assumed
a constant
distance
between
the
first
Reynolds separation
transition, UJ: Rt, --
w
As the forces a value
Reynolds the
laminar
of 40,000.
number length This
increases, to decrease expression
in size. can
-
w
-- 50,000
Us usually
el
z
V
increases, Horton
be rearranged
[40,000
(4.1)
(6 )s
too, [1967],
such thirty
that years
this
criterion
later,
used
as
(4.2)
44 w
O'Meara
and
Mueller
[1986]
took
an experimental
fl
which
is a less general
these m
relationships
reported
value
[Crabtree,
aration
of about
750,
and
Mueller
describe
able
to capture
data
sets
this
used
which
how
the
precedes
correlation
by allowing
by O'Meara
and
thickness
transition
of data
as
Rtl.
Schmidt
and
[1989],
experimentally
Reynolds
by
a variable
Mueller
at the
laminar
developed
Mueller,
and
of the bubble
of momentum
over
effect
out by Schmidt
the vanishing
1957]
sets
(4.3)
As pointed
capture
many
C
relationship.
cannot
from
_ 155(52)s
c
m
mean
number
at sep-
separation.
Vincent
de Paul
Similarly, Mueller
Schmidt
based
[1972]
on the
propose
the
is
same
following
criterion, gl
such
that
this
the
criterion
onto
While sition w
is reflected
[ [267 -
0.3709(R_)s]
c , (_2)s C '
72 < (R62)s
< 720
length
bubbles,
vanishes
form
dependence the
and
the
coincidence.
J
< 72
that
therefore
better For
flows,
Larger
short.
bubbles
that
occur
longer.
Fig.
4-2
far
on large values
downstream
illustrates
this
coarse
values peaks
of the
are
usually
stagnation
where
the
the
this
traneffect
correlations thickness
due
to a spurious
correspond
to leading-
associated
isolated
The
stagnation
point
origin.
on the
bubbles,
bubbles
not
whose
the
influence
shows
does
of momentum
of (52)s near
4-1
(4.3)
through
way.
value
Fig.
as a plane
of separation
to the
of (52)s
line
(4.4)
Eq.
figure
a strong
of leading-edge
suction
Since
in the
has
length,
small
effect,
(4.3).
in a rather
length
> 720.
a straight
number
overall
in fact,
Eq.
is simply
is proportional the
of (Re2)s
it is shown
correlations
predict
form
quite
on the
length
which
usually
plane Reynolds
of these
airfoil
with
on (R62)s,
the
transition
at values
together
(52)s/c-g_/c
in many
at separation
edge
27 < (Re2)s
it is true
length,
for which
(_2)s
bubble
any
projection
3.7820(Re2)s]
in graphical
incorporate
m
_
c
u
f [513-
and points
point with are are
and
are
mid-chord
usually
much
leading-edge
45 w
SCHMIDT
0.I0
w
O'MEARA
AND
MUELLER
0 72.12
301 0.001
(6 )S .Z
•720
C
Fig.
4-1
Comparison for the
of two
separated
empirically
laminar
shear
derived layer.
transition
correlations
m
f-
46
;
i
C
0.20
0.10
w
72.12
301 0.001 463
72O
4-
O'Meara
D
Schmidt
and
Muelle:
w
L
_
Fig.
w
4-2
Comparison
of the
laminar
correlations
for the
Eppler
lengths E387
predicted airfoil
by the
two
at R = 300,000.
47
bubbles.
The
about
the
so far
no local
bubble
of (_2)s,
stability
developed ically,
value
with
of the
can
however,
separating
capture
criterion
the
can
increasing
effects
capture
angle
location tend
success
in attached
such
above at
the
seek
based bubble
local
models,
of separation, crement to better
explain
development
plot
Integration _2 and
known.
layer
edge
downstream =
these
two
w
variables, point
at
a value
location
of the
boundary
the
layer
that
empirical
criterion
distribution.
Specif-
upper-surface
mid-chord
criterion
the
it should
be possible
that
Whereas free
the
shear
the
here
the
in predicting
development
accurate
at
also
each
transition
is implemented,
and
conditions criterion
unlike
is calculated
in the
is early
downstream
downstream
prediction. Eppler's
to ex-
described
Eppler's
Since,
transition
criteria
layer
airfoil,
layer.
development
to a more
Eppler's
on
boundary
monitoring
lead
in-
In order
boundary-layer
be described. momentum
inviscid
that
station
subsequent
occurs
is at a different
downstream
velocity
nects
stagnation
in the
of the
Since
it was thought
transition
should
_3 at each
also
layers,
criterion
between
how
information
Criterion
layers.
naturally
any
no
pressure of the
boundary
it seemed
would
Transition transition
characteristics
the
In fact,
vanishing
to separated
which
on the
layer.
contain
of attack.
boundary
a correlation
separation,
in itself
of a variable
the
of Eppler's
a criterion
not
shear
Eppler's Given
does
can
drives
and
The
shape
velocity
along
the
the
(1-132,R62)-pairs
to the of/-/32
boundary-layer to obtain on
describing
trailing =
integral
station.
be grouped
thereby
energy
edge 1.62
and
a plot the
factor airfoil
H32
gives =
is taken
development, the
whose
development axes
concise
approximately
way.
the
values
of
_3/¢52
is therefore
as the
boundary-
U and
62 at each
of R62.
measure
boundary-layer
in a very R6_
equations
Eppler
con-
the
variation
in
development
from
the
The equal
stagnation to 10.
point Values
of
48 R_ 2 < 10 are
not
Following
upper
tion
the
dotted
until
bubble
mately
correlating
trend
since
the
nar
parameters matched
at transition
dependence > 875,
/-/32 to increase
turbulent
/-/32 = 1.51
(H32):r
with of very
growth
dependent
to 1.51.
toward
for smaller
does
as P and
is weakly
edge
Thus,
again
The
causes
of these
as shown generality.
local
attached the
flat-plate
which
investigations,
part
of
approxi-
of cubits
with trend
intro-
to disappear decreasing
in laminar
distribution
this
similar
length
in the
criteria
P.
lami-
served
to
bubble
conditions
incorporated
in
any one
particular
bubble
be
figure,
again
fully
again
transition
is seen
any
to a value
thickness
gradient
it was found
criterion
value
for
in the that
the
Eppler's
curve
tur-
number
of 1.77.
the
next
was
the
development
drives
at H32 = 1.46
could
at reattachment
Reynolds
developed
will be discussed
such
of transition,
boundary-layer
pressure
occurs
particular
bubble
as a
be constructed
pressure
in the
momentum
plot
laminar
as a family
and
the
R62 stays
The
a similar
separa-
growing
therefore
Downstream
turbulent
adverse
which
method,
while
H32 to decrease
on the
separation
for Drela's course
limited
the
to local
In fact,
a criterion
toward
is approached,
on the
unsuccessful,
(Re,)s.
the
(H32)7-increases
imply
correlates
the
on
inside
can
figure
while
values
Although
easily
trailing
on P.
laminar
starts
separation,
(4.4)
in this
line,
H32
to (R62)s.
shown
bubble.
layer
close
layers
depends
bulent
In the
shear
of H32
found
always
L-i
of Ha2
laminar
development.
is shown
point,
to Eq.
of growth
consistently
which
from
necessarily
such
shear
distance
solid
monotonically
value
poorly
quite
Ha2 grows
the
criterion
separation
not
of the how
laminar
boundary-layer
by
This
criterion
in (Ha2):r
illustrate
given
similar
of (R62)s
part
a typical
A very the
rate
the
downstream
an additional
This
shows
Ha2 = 1.515095.
While
for separated
at values
4-3
development
From
is met.
constant.
duces
r_
line.
with
criterion
L--
when
transition
the
Fig.
surface
is encountered
vertical
by
plotted.
boundary
and causes
As
the layer
method chapter. corresponding
and
49
@
D'I
@
w
@
L_
_k
L
,-_
ORIGINAL
PAGE
fS
OF POOR
QUALITY
"_
50 to attached where
transition
a bubble
number
be higher transition
aration
line
modified
at
the
This
transition
precedes
laminar
the
value
criterion
[Eppler,
Re 2 = 463.
criterion
Based
now
sition
is still
order
This
leads
This
apparent
mainly _=
predicted
million
transition
too
to a value
when
bubbles
at the
the
value
and
high.
Mueller. the
measurements,
cases
of Reynolds
be quite
intersected
coefficient
In fact, Eppler's
laminar
Eppler
sep-
[1989]
has
The
reason
numbers
why
chord
Reynolds
cause
any
significant
drag
increase;
thickness
that
is given
is due
rather
than
transition
is simply
that
in fact, by the
separation.
Reynolds
is replaced seems
to its having from
such
it usually attached
is near
which
tranIn
numbers with
190.
to work
well.
been
calibrated
a modelling
of the
instead
of a bubble
is predicted at
however,
(4.5)
= 875,
(4.5)
a value,
at chord
125 in Eq.
criterion
1.573) 2
layer
on airfoils
of ]_
data
-
For such
boundary
the
in Eppler's
drag
+ 125(H32
forming
intersection
high
momentum
shown,
additional
approximately,
coarseness
process.
not
at R62 = 704.
soon
to compare
airfoil
can
by Schmidt
+ 18.4Ha2
occurs
or greater,
from
the
in many
to
intersection
to be able
of one
_
the
because
separation
1963], on
to be predicted
happens
of 720 proposed
[in R_2]_- _> -21.74
where
transition
present.
than
original
natural
is actually
at which
it can
will cause
values
the
bubble
causes
the
same
turbulent
boundary
actual at
does
not
increase
in
layer
over
very
suc-
together
in
its length. In an attempt cessful Fig. w
transition 4-4.
to observed pler's ]:t62 vs.
In
to reconcile prediction
the
transition
boundary-layer 1-112 in order
Eppler's method,
e n method,
transition the
a value in many
development
plot,
to include
several
e" method, of r_ =
locations
criterion
different this
they
9 has
are
been
flows.
comparison
measurements
with
another plotted
found Rather
is done of these
to than here
variables
correspond using
Ep-
on a plot at observed
of
._
o_
\
-_v. U
e,I
o
o
0_
"_.,
_
_._
o
I
© © L _
_..I
el
I_
52
transition locations. This comparison method
was attempted
also because, since the en
can predict transitionsuccessfullyin a separated shear layer,it was hoped
that a suitable criterionfor use in the boundary-layer development
plot could be
inferred from it. In this figure,contours of constant n are given for the FalknerSkan self-similardevelopments
[Schlichting,1979]. For such developments, in fact,
a unique surface existsthat allows the determination of the value of n from the local ;
z
:
A
values of the momentum
thickness Reynolds
number
and shape factor alone. The
equation defining this surface was developed by Drela [1986] and will be discussed below. Shown
in this figure is Eppler's transition criterionfor attached boundary
layers as well as one of the attempted
criteriafor separated flow. It is interesting
that Eppler's curve fallsquite close to the n = 9 contour, for zero-pressure gradient flow (H12 = 2.59). As can be seen from the wide scatter in the experimental data, however, it seems dubious that a single curve could capture the correct transition location inside the bubble with any generality. In fact, the growth of n in a nonsimilar boundary shown
layer development
will not follow the surface whose
in the figure,regardless of whether
contours are
the flow is attached or separated. This
implies that transition is not dependent
solelyon the local boundary-layer charac-
teristics but depends also on the manner
in which the boundary
=
_=_
layer arrivesat such
values. The inclusion of path-dependency, or the effectof upstream development
boundary-layer
on transition,has captured the generality that the transition criteria
discussed so far lack In order to fully understand
why, the en method
will now be
discussed in detail.
The M
ago
The
e n semi-empirical
[van
In_;en,
to a variety
1956;
Smith
of aerodynamic
transition and flows,
e n Method prediction
Gamberoni, relies
method, 1956]
on a theory
developed
and
since
that
can
thirty
successfully explain
the
years applied
moderate
53 successof someof the correlations described above,that can correctly distinguish betweenleading-edgeand mid-chord bubbles, and that can correctly model the effccts of variations in pressuredistribution upstream of the bubble. Linear stability theory, in fact, directly models the growth of instabilities in a boundary layer while indirectly, through the boundary-layer development, accounting for the effects of Reynolds number, plitude take
at station place
when
transition to lie number
n reaches
9 by many
It is generally process
and
fully
flow
between 1984],
process
is usually
employed
be completed
in the
of the
the
use
the
e '_ method
actual
last
nonlinear
n = 9 -
14 and
the
amplitude
of the
method
oncoming
empirical remains
the
the
spots
usable.
to model surface
disturbance
conditions
corrections
[Mack,
engineering
transition
fully
are
stability
influence
of the
In spite
prediction
is taken
input
with
this
is unknown. turbulence
of transition of these
method
to
abruptly
associated
onset
the
empirical
at neutral
1977].
turbulent
started
significant
on the
defined
function
transition
concern
the
transi(n = 0)
intermittency
often,
cases,
Another
directly
an
the
to approximate
calculations
In both
Reynolds
is usually and
In order
region,
turbulent
on
stability
however,
to n = 8. Equally
of the
air and
distance.
reported
models
neutral
of turbulent
station.
method
correctly
region,"
been
1990].
between
amplification
streamwise the
theory
has
to
observed
to depend
et al.,
distance
of this
close
value
am-
is assumed
to experimentally
it appears
stability
of disturbance
Transition
this
Horstmann
appearance
ratio
so.
correlated
"transition
30%
at a value
to render
of further
linear
first
stability,
although
70% of the
The
of the
environments
1989;
that
or the
when
is that
inability
tensity
flow
the
corresponding
is necessary
The
similar
accepted
transition
method
previously
Vemuru,
flow.
logarithm
at neutral
researchers,
and
turbulent
[Arnal,
as the
a value
for approximately
region
at the
For
[Evangelista
tion
is defined
s to its amplitude
locations.
around
as the
n(s)
inforces
weaknesses, most
faithful
54
to the
actual
The
cn
equation
physical
process.
method
is based
which
dimensional
is derived
from
Navier-Stokes
of locally
parallel
mean
partial
differential
ficients
are
particular
the
the
numerical
are
flow is made.
only
solution
Navier-Stokes
equations
equations
functions
on
equations
perturbed
Subtracting
for the
method
and the
perturbation
of y, the
field
An
The
a system
is obtained.
Since
can
two-
assumption
flow,
modes
F
implicitly
y-direction. and
as follows.
linearized.
mean
of normal
Orr-Sommerfeld
of three the
be applied
coefin the
form
v'(x, which
of the
Similar
co* the
reference
assumes
length
a sinusoidal
expressions
radian
frequency, and
"1
=ae p(y)e
(4.6)
disturbance.
are
assumed
which
can
Here
for u I and
v I is the
disturbance
pl.
a* is the
be nondimensionalized
with
in the
wavenumber respect
to a
velocity, o_ = o_*Lref co*L_,/
(4.7)
co--
Substitution turbance
of these field
turbances.
expressions
results
in a set
Eliminating
into
the
of three
t__ and
partial
ordinary
/31 in favor
differential differential
of 9t
the
equations
for
the
dis-
equations
for
the
dis-
Orr-Sommerfeld
equation
is
obtained,
5,1,,+ [_in(_u, a are
and
¢o are,
performed
obtained Two
_),
in general, in the
by taking limiting
the cases
2_2]¢, + [iR(_u _ co)_2 +inu,, both
complex real
complex. plane,
part
of interest
Although it is understood
of any
of the
complex
derivation that
and
physical
variables
the
(4.8) analysis
quantities
are
employed.
are
{ o_ co = Wrco += icoi a = = o_r, a_ + iai, w_
w
the
+ _4]_ = 0
temporal instability spatial instability
(4.9)
55 To calculate the amplification of disturbances as the boundary layer develops, a spatial instability analysis is necessary.Thus, the Reynolds number and the frequency are specified and the wavenumberis found by solving numerically the OrrSommerfeld equation subject to the boundary conditions dv t v 1-
The
homogeneous
or eigenvalues -ai
boundary a.
is largest.
-0
dy
The
This
value
at
y=0
and
conditions
imply
of greatest
physical
can be seen
from
the
v' = _(y)e
as
y_ec
the existence interest
expression
(4.10)
of an infinity is the
for the
of solutions
eigenvalue
for which
fluctuation,
i[(_7 +ia,'. )_-w: tl
(4.11) = where the
_(y)
is the
bracketed
term
reference
point.
represents
the
layer
distribution
The
is the
amplitude
ratio
between
amplification
developing
of the
in zero
disturbance of the
this
term
amplification
factor,
amplification
In this the such
case,
absence
this
factor
-2
v( )e
n, is defined
curve
turbance
gradient
them.
Assuming
x from
some
x-locations a boundary
__;(,2_,,)
as the
(4.12)
logarithm
of this
ratio,
• equal
is a constant
of a pressure
as, for instance, The
is thus
at two different
and
I
v(y)e-,_,
o= The
at a distance
evaluated between
(eigenfunction)
gradient, ^
A2
The
disturbance
of a disturbance pressure
amplitude
to the
area
because
prevent
the
any
(4.13)
under
the
amplification
parallel
change
flow
in the
rate
curve.
approximation
local
Reynolds
and number
R62.
continuous
wavelength
travelling
downstream
modulation
encountered
in a developing
boundary
by
a fixed-frequency layer
is approximated
dis-
56 locally by a
series
Sommerfeld
equation
and
at different
and
H12.
plateaus,
for a sequence
pressure
As the
generalized
of constant-wavelength
of parallel
gradients,
streamwise
each
extent
obtained
mean
flows
characterizable
of the
plateaus
by
solving
of different
to zero,
value
Eq.
Orr-
thicknesses
by a different
tends
the
of R62
(4.13)
can
be
to ?2 _
--O_ i*
_2
dx
(4.14)
1
As
a measure
of how
calculate
the
the
branch
lower
total
far
from
transition
amplification of the
that
neutral
a boundary
has
occurred:
layer
the
is,
it is preferred
amplification,
that
to
is, from
curve,
=
(4.15)
o_$5
0
where front
the
independent
stagnation In general,
not
The
local
Reynolds
with
time,
with
the L _
may
the
come
layer
environment
close
distance
that
the
the
but,
along
the
in simpler
airfoil
from
of several
n = 9 indicating
it may
not
upstream
to n = 9 upstream
separated
of separation
of broad-band that,
the
does
or white at
a specific
dynamical
systems
is called
Reynolds
number
changes
are amplified
as the
different
frequencies
boundary
layer
at the
same
transition. shear
be necessary
of separation.
or in free flight
frequencies
boundary-layer
frequencies
in the
tunnel
rather,
to what local
amplified
layer
pulse
those
tracking
reaches
in a wind
amplifies
different
of separation,
boundary
simply
Since
necessitates first
pressure
correspond
distance,
frequencies
upstream laminar
disturbance
frequency.
downstream
As the those
the
number,
This
s, the
point.
boundary
natural
develops.
is again
of a single-frequency
noise.
the
m
consist
variable
The does
layer
may
be ditTerent
to monitor fact not
that
the some
necessarily
from
stability
of
frequencies imply
that
w
transition
v
will
occur
sooner
once
the
boundary
layer
separates.
If this
is true,
57
then
it should
information,
be
for instance
of all previous this
possible
such
to
devise
a transition
on conditions
criteria
would
criterion
at laminar
still not
based
separation.
be sufficient
solely
The
proof
on
consistent
of the
local failure
untenability
of
hypothesis.
Drela's While
the
questions
approximation
to the
in the
Eppler
duces
an error
it is now
and
Rather
above
e '_ method
in the
than
raised
Somers
discussed
Approximate need
further
developed
program.
study,
by Drela
In order
calculation
in some
e" Method for the
[1986]
has
to demonstrate
of n over
and
above
present been
that
time
implemented
this method
its declared
the
intro-
approximations,
detail.
performing
a linear
distribution
as can be obtained,
downstream
station,
the
characteristics
stability
analysis
for instance,
following
Gleyzes
from
et al.
of the boundary-layer
a finite-difference
[1983]
Drela
velocity
method
computes
at each
a data
base
of
L_. stability
a boundary-layer More
precisely,
value
of the
number
of the
calculation the
Falkner-Skan
using
the
nondimensional
local
shape
is divided
by
the
local
growth
factor
profiles shape rate,
of a Falkner-Skan
local
boundary-layer
that
factor -a
can
be "tapped"
as the
coupling
i, corresponding
profile
and
during
parameter.
to a particular
of the
characteristic
local
Reynolds
thickness,
i.e.
62,
=
to obtain the
the
correct
integral
physical
growth
characteristic
equation,
the
to be used
thickness manner
is of no consequence. self-similar
rate
The
developments
is obtained
in which
most
at constant
integral
independently,
the
convenient
in the
from
nondimensional way
H12 values
(4.15).
data
is to calculate and
increasing
the base
the
Given
momentum is generated
growth
R6=.
that
rates
Starting
for from
L m
the
Orr-Sommerfeld
different
values
spatial of/-/12,
instability
a set
analysis
of neutral
curves
of the
Falkner-Skan
is generated,
one
profiles for
each
at many value
L
shape
factor.
An example
of these
neutral
curves
is shown
in Fig.
4-5 for two values
of
58 of H12.
For
is evaluated
each
value
of shape
along
rays
of constant
factor,
the
reduced F-
to form
curves
development
-o_i(H12,
of each
R52,
F).
dimensionless
rate,
these
profile
(4.16) curves
is found
n( HI_ , Rs_, F) =
-ai,
frequency, 2rcfu U2
From
Falkner-Skan
amplification
jr8
the
amplification
factor
for the
from
-o_ ds
$ 0
r% _aT dR_ = fJ R6_o ds For
a Falkner-Skan
profile,
u where •
rn is constant
(4.17)
and
-
u
defined
1+
m U
f'(1
-
f')dq
as
t
s dU m
(4.19)
--
U ds Denoting
the
(4.18)
momentum-thickness
integral
(not
a function
of s) by I,
Z
R_
=
1 -Fm
u
I
(4.20)
w
Thus,
v
=gI
l +mus
__m
=I
l+m_,s
--
[(l+rnU)½_] _s
1+
---_s U
2
__= !
..
=
I22
12 62
(4.21)
59
Thus,
Equation
(4.17)
becomes,
n( H12 , R_2, F)
-(Wh)
1 fR6_ = -fi ,j r_,2° 1
f_6_
-[Z(H12)p J_,20 In
this
way,
Sommerfeld
wave
dR62/ds The
H12
Gleyzes surface
dimensionless
can
number assumes
curves
are
original
equation
mensional since
the
be and
the
obtained
shown
in Fig.
et al.
[1983].
for selSsimilar
used distance.
with
the
frequencies
form
this integral Drela
takes
This
leads
to the
of
which can
F)
dR_
the
dimensionless
is defined
be done
(4.22)
for
in
terms
Orrof a di-
a self-similar
profile
shown. for different
the
frequencies
envelope
following
and
as a straight expression
at a constant
line
for the
as done
by
amplification
developments,
dn
n(R_,H_)= where
n,
This
4-6.
-ai(H12,
eigenvalues
to find
particular
d/G
superscript
"e"
H
d--_ (_)
denotes
a value
[_-R_0(H_)] obtained
from
the
(4.23) envelope
of amplified
and
_(dn
H
_)
]
e = 0.01[{2.4H_2
- 3.7
+ 2.5 tanh[1.5(H_2
l°gl°[R62°(H'_)]
=
[[H_2:1 1.415
0.489]tanh[
3.295
+
-
3.1)]} 2 + 0.25]½
20 1 H12-:--
+ 0.440
(4.24)
12.9] (4.25)
H12 -- 1 ----:2
While
Gleyzes
et al.
then
evaluate
the
amplification
dn R, 2 o
H
integral
as
(4.26)
6O Drela
changes
the
variable
of integration
back
to s,
1• dR62
_(_)= j[ s [--(HI_), dn o [dn_,
At
this
point,
Drela
finds
d-'_
-
an expression
u
_
ds
for dR_2/ds
1 s dU U52
U d52 +----+ u ds
us
shows
that
ond
term
inside
differential
dR_2/ds the
dU
equation
equals
square
us
in a rather
roundabout
way.
1 52 dU
us
this
d52
bracket.
152
d--7+ _-2_
last
expression
If this
can be integrated
:.
(4.27)
2 u ds
=_ L_-_2+2_, He
d_
+----u ds
2 U ds 1 [s
j
with
is taken
_ a
_
1 in place
as a condition,
of the the
sec-
resulting
as follows,
(
U d52
--
1 5_
2 _--_-_2 _ d---_- + 2 u s d52
s dU
26_ d--T+ u d_ - 1 s
[ 2
lnS_)
+ _s(lnU
= 1
d
= 1
_
[ln(5_U)]
ds = --
d[ln(5_Y)]
3
ln(52U)
= Ins
+ C
5_U = Cs
52 = CIu
(4.29)
r
Thus,
the functional
the
constant
two
"empirical
form
is in general relations"
characteristic a function (given
of Falkner-Skan of the
in the
shape
factor.
profiles Drela
is recovered, now
introduces
where the
Appendix),
s dU U ds
u_ 118 u
-
re(H12)
-e(H_)
(4.30)
(4.31)
61
\'Vhat he
he means
is using
these
assumption
profiles
correlations
here
and
of a Falkner-Skan
substitute dR_2/ds
is "analytical
the , Eq.
Falkner-Skan
from
since
his
62-growth,
it
expression
for
the
form
Falkner-Skan
for
is not 62(s)
profiles."
dti62/ds
clear
why
directly
into
Since
indeed
implies
he
not
did
his
the
simply
expression
for
(4.28),
ds
-
2
_
+ 1
us
62
o_
1 +m
]2
62
IS (4.32)
62
which
is identical Having
to Eq.
established
(4.21).
Thus,
integral
only
had
to curve-fit
I(H12).
that
re(E12)
Drela's
Drela
for evaluating
2
+ lt(g12)
= [i(//12)]2
(H_2)
2
(4.33)
n(s),
n(s)=
62(s)
ds
(4.34)
0
.
°
can
be written,
using
Eqs.
(4.17),
(4.21),
(4.26),
and
(4.33),
(4.35)
"r;
L
where
62(s)
comes
by the
momentum
from
the
and
energy
non-similar integral
_-2
J
6-_ds
boundary-layer equations.
development
as calculated
Thus,
W
n(s)
m w
= fy " [-a*62]¢ds
(4.36)
=
(4.37)
o
o _
ds
62 Which
is identical
of an envelope precisely n.
from
to find
what
That
to the expression
enables
oq.
Using
the
method
is, the dimensionless
the
value
thickness at the
of H12
which same
52(s)
and
R,2
the
actual
straight the
envelope line,
is divided
by
from
known
to rely analytically
fictitious
two
following
that the
obtained three
(4.22)
the
reduced
Fig. factor
with 4-7
the
shows
occurring
The
three
the
argument.
along
momentum development
It is desired
shown
envelopes
jump
Fig.
4-6
given
n-growth
along
the
the
to the airfoil
Thus,
the
in H12 shows
by
curves
Eqs. the
the
error it is
are
not
to envision
a
with
corresponding
a to
of n obtained
at constant
frequency
to
curves
for
amplification plot
of
this,
lengths
growth
as calculated
(4.23)-(4.25) boundary
The
in question
the
a
envelope
To see
curves
used
it with
true
it is helpful
neutral
neutral
had
surface.
functions
to compare
the
on the
if Drela
up of two constant-H12
4-5 shows
envelope.
even
is non-similar.
since
made
through
the
of
station
of approximating
correspond
numerically.
Fig.
factor.
frequencies
not
that,
instead
development
development
Drela's
growth
downstream
in a self-similar
found
curves
frequency
obtained
development
value
is
Error
example,
in between.
using
together
of H12. shape
are
boundary-layer
use
equations
on the
the
author
would
a numerical
of shape
the
amplification
boundary-layer
but
jump
values
[1990],
at constant
boundary-layer
discontinuous the
on
history at each
local
for the
governing
obtained
the
Envelope
n(s)-development
the
by the
except
R6_.
Selig
of the
curves
whenever
necessary
date
his
amplification
arises
by
with
Degenhart
for upstream
-o_i
The In collaboration
and
as calculated
rate
different
of H12 and
by Stock
to account
growth
is in general
value
used
for
the
by Eq.
two
layer
with
the
that
serve
best
values
switch
in
at R_2 = 500.
frequencies
selected
Referring
represent
to Fig.
4-7,
limiting
cases
as /_6_ increases
n(F1)
grows
to eluciaccording
63
b.
0 0
i I I I
! I ! I I
N
0
k.
\
I ! I I I I I I ! I I I I I I ! I
,
I I ! I I ! I
D.,
0 0 0
_J
0 u3
N 'O
n,'
0
m
0 0 In
2;
I 00 0 0
I
I
0
64
0 u
0
N ',O
t_ ;> 0
u 0 u
w
O
3: 0 ..J
_4
0
/11:
I a3 r_
U'I
• 0
II ,..I
II
I
N
I m
...I
oo
y
_
0
0
m
c_
l"....
0
i° II (
....
t
....
J
....
J
....
J
LJ
_
m
&
_
96
o. .C_ o" 0 0
_z
R
o_
0 0
X °_
°_
tn 0
/011mz-J
0 L_
O
ml-o
o C_
!0
0
_b c_
0 0
o
o
Co
97
of the
amplification Fig.
6-1(b)
analysis. the
Reynolds
this
plot.
as the
total
drag
lower r
=
This and
come plot
point.
figure
serves
using
the
compared
with
very
with
ttl_
vs.
plot
shows
original
separation
give
locations
the
z/c.
similar
and
without
At the that
point.
bottom
laminar
of z/c,
turbulent
Goldstein's of the
this
the
singularity
can
and
the
upper
and
is shown all
five
denoting be
clearly
point. airfoil
boundary-layer
is symmetrical
surface
method
bubble
difference can
the
analysis
of the
airfoil
as
Finally,
boundary-layer
Eppler's
chord,
plot
asterisk
lower
The
extents
lift
the
the
singularity
for
with
the
exception
figure
on
process.
Since on
as expected.
separation
design
points.
the
to the The
and
a bubble the
development
separation
turbulent
With
left
lengths
the
Drela's
results,
bubble
on
Eppler
that
respect
the
model
the
includes
surfaces.
Goldstein
bubble
Eppler
with
of the
several
This
lower
as functions
through
and
n = 9.
the
by
with
is drawn
indicating
and
during
together
bubble out
label
boundary-layer
removal
the
model
and
Eppler
to illustrate
prediction
methods
next,
developments
at
upper
useful
The
suppressing
laminar
printed
the
boundary-layer
for reference
calculated
normalized
the
are plotted
0, by
the
from
is especially
variables
transition
of the
as printed
below
lengths
The
The
analysis
right,
with
outputs
is provided
viscous
arc
last.
and
of attack.
to the
are
smooth
c_ =
of the
coeMcients
coefficients
in the
angle
separated
drag
This
distribution
performed
reattachment
seen
inputs
is printed
boundary-layer the
relevant
summary
surfaces
below.
all the
and
and
well
separation.
number
been
of turbulent
laminar
velocity
program has
after
inviscid
The
analysis
!
contains
The
Somers
occurs
be
in separation from
boundary-layer occurs
at
can
be
starting
region,
seen
present
the
at
the
two point
plot
of
development
the
boundary
with
m
natural analysis,
transition however,
according indicates
to the that
modified
n at separation
transition is still
criterion quite
Eq. small.
(4.5). This
The apparent
en
98 inconsistency
of Eppler's
calibrated
using
surfaces
derived
As shown
The
more
separation.
In fact, appears
laminar
part
a few percent
as will
of the
chord
part
shown
in Fig.
start
before
any
data
the
by
in the
pressure
This
airfoil
was
Tunnel.
Drag
polars
for
and
drag,
also
R = 2,000,000, on
effects
low-drag
of low-drag data
Fig.
high
of the
(top
500,000
the
at the
(bottom
from
bucket).
since
6-3 shows
here
and RPV
design
effects
the aerodynamic
and
present of the
at
laminar
a mid-chord thickness
transition
plateau
in
point
in the
transition
may
indeed [1978].
Low-Turbulence for
lift,
The
is therefore It was also
was
designed
predictions bubble
characteristics
are
for
surveys
obtained
designed
characterized
of R = 2,000,000 R = 700,000
wake
were
profile
Pressure
at
for use
by
a large
to minimize
the
for
a high-speed
maximum
endurance
are
compared
most
clearly
with
the
R
L5
_=::=
by the original
Eppler
and
Somers
program.
=
seen.
of this
airfoil
on the
Shown
is the original
plot
as gen-
w
erated
is
laminar
by Russell
measurements
conditions
bucket)
to an increase
length
as maintained
Langley
500,000. and
of the
measurements
ct requirement.
The the
NASA
force
interpretation.
assumed
calculated
the
Airfoil
distribution
700,000,
bubbles
dash
in the
long-endurance
to achieve
of the
tested
inside
in momentum
experimental,
is observed,
pressure
1,000,000,
The
lower
deterioration
lead
been
growth
NLF(1)-1015
calculated
detailed
a high-altitude
aft-loading
w
recently
to the
this
a certain
As the length
close
NASA
The
any
it was
and
thickness
support
had
that
upper
to cause
necessarily
below
evident.
too far downstream.
change
not
if transition
is clearly
is quite
may
drag.
of the
of momentum [1980],
later,
by realizing
drag
not seem
Roberts
shown
airfoil
bubble
6-1(a)
does
the
values
than
be
be resolved
case,
6-2, a bubble
greater
to reduce
this
experimental
in Fig.
thickness
For
may
the bubble
Gault's
clearly
criterion
data.
Therefore,
from
in momentum
bubble
drag
performance.
bubble,
the
mainly
is identical.
in airfoil
transition
prediction
99
_ _c_ "_ _o
o
-
w
"\o
_ _.a _
I \ \\
I
o_
i
_ o"__,..._.._:_.
×
/ o/
!
II
r--
-< o..n ,_ -I.t_ o_
LL
I I *
I i I
_,ilil]i]_
....
I
....
I
'
100
obtained
by assuming
model,
Drela's
difference is due
transition
XFOIL
in drag
results,
In fact,
bucket transition
bubble at laminar
produce
the
of the
polar,
the
the
to a drag
data
onset
original
data
from
program
and
methods
reduction case,
excellent
of strong
prediction
global
of the present
LTPT. the
how
is calculated
the
present
formulation
interaction
upper
cannot
be
version
at the
what
At the
of the
as shown
over
accuracy.
Part
present
employed,
6-3, it can be inferred
In any
with
the
experimental
2-4 and
separation.
measured
separation,
boundary-layer
Figs.
leads
the
between
turbulent
comparing
the
and
prediction
to the different
2-4.
at laminar
in Fig.
top
by
of the
assuming
is able and
to re-
lower
neglected.
limits
At
these
m
=
conditions,
the
the
moment
lift and
separation the
W
the
z/c
program
tween
the
transition It can
as the
pressure
adverse.
This
of transition they
occur
also
be
for
curves how
the
given
as calculated
two analyses
represents
decreases
upstream
of laminar
present
plots.
:r/c, model.
the
they
laminar
becomes
appear
the
to occur
As will be shown
more the
version
of
difference length
eventually
Because
labels
between
point,
and
turbulent
a rather
original
the
contains
program
the
separation
in subsequent of actual
by the
in size
the
difference
Since
separation
also and
s_,,,.b/c,"
little
laminar
bubble
in terms
very
plot
transition
"1 -
of presentation.
shown
The
although
is really
is usually
at the
for the
will be clearly are
there
ease
transition
distribution
beof the
disappears increasingly XFOIL
tran-
downstream
below,
however,
upstream.
Matching prediction.
is used
poorer.
In actuality,
variable
that
are
of c_ as well as the
of x/c.
independent
be seen
locations
predictions
as functions
Given
assumes
bubble.
sition
the
variable.
variables,
drag
as functions
as "z/c,"
cumbersome
lift and curves
locations
axis
two
present
the The
compared
experimental pressure
drag
distribution
to experimental
polar and data.
does the
not
by itself
boundary-layer
Unfortunately,
guarantee
an accurate
development it is much
more
should difficult
101 to measure these quantities. Consequently,few data sets are available. For the NLF(1)-1015 airfoil, the details of the bubble flowfield can be checkedonly through the pressuredistribution, since no boundary-layer data were taken. Figs. 6-4 to 6-6 show typical comparisonswith the measuredpressure distributions together with the corresponding calculated developments. Fig. 6-4(a) shows the pressure distribution corresponding to the lowest value of XFOIL
prediction.
on the
upper
bubbles
upper
and
surface
lower.
to see
how
bubbles. of the
surface
very
different
the
other
are well approximated
on the the
Two
Fig.
6-4(b)
large
the
These, airfoil.
mid-chord
and
The
a very
the
case,
do not
XFOIL
seem
together
with
at the
mid-chord
one
on the
lower
shorter
on the
pressure
distribution
can
become
inside
sin(l/x)
function
be clearly
for CD,,,,,
are very
similar,
bubble on
It is interesting
much
can
the
Both
a slightly
development.
factors
surface.
gives
to contribute
of the values
edge
perturbation
shape
polar
be seen,
boundary-layer
of the
The
can
leading
model.
slight
values
shape
bubble.
at the
by the
shows
in any
bubbles
ct on the
leading-edge
to increasing
the
the
recognized
greater
drag in the
increase
in 52
of the upper-surface bubble being largely due to the longer extent of itsturbulent part.
This
edge
bubble.
variable
bubble, The
has
little
in fact,
is much
distribution impact
thicker
of cy seems on
the
at the
transition
plausible.
boundary-layer
In any
point
than
case,
in this
development
the
or on
leading-
region
the
this
pressure
distribution. Fig.
6-5(a)
to the
middle
along
the
mental. by
the
shows of the
bubble, It can
the airfoil
the
be seen,
large
aft-loading
simultaneously.
In any
pressure
distribution
polar.
In an attempt
inviscid
angle
of attack
however,
that
the
prevents
the
matching
case,
the
model
strong
reproduces
at an angle
of attack
at matching is one
degree
the less
trailing-edge of the the
gradients measured
corresponding
pressure than
gradient the
experi-
interaction
induced
on
surfaces
both
pressure
distribu-
102
C_ O C_ o
O
×
X
--_ _._...+,a
c_ 00
o Z_
/
m
I°
'I......... i......... I......... i.......
•
o
CD I
I
k
o
CD I
g
_ I
.o C_ 0
'"I......... [......... I......... I......... I......... I' CD
_
(_
CD
CD
,-]
_ I
cD I
CD I
CD i
•
CD I
O
O I
Q_ .< C_
o
I II _
x -Tr_ ._J _C
0
0 II
LLJ = =
i •
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0
m
103 0
\
" \\
o
0 °"_
O
va_
I ¢.9
_o'
w
/
!
m ,¢...-.
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, /
II
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o
@
!
o 0 O
O
O ¢o
,5 11 B
0_
•
v
W
w
I
O [
I Q..
0
i3
107
/ m
_
P
_
%.
"-
j
"f
I" I I I I
/
I I I •
•
J
•
,
a
J
i
I
,,,I
i
CO
....
I,,,
0
2,1,-,
-r-
l
= J
II fl II tt
i,J
°°
/
W
I
....
I
III"
/ y I ....
f l I
I
....
I
....
II
]
108 tion
remarkably
may
be
expected
to be thick.
tion
the
drag
to the
a little
well.
due
short
starting
although
turbulent
surface
6-6(a), bubble
total
a fifth
proportions. At
the
laminar
the
separation
leads
As far
as the
a destabilizing
of the
long
at this
and
condi-
by XFOIL
are
this
of attack,
n is
angle
where
point.
the
upper the
surface.
step
The
in _2 in the
how
the
correct the
until
transition
on transition
are
the
the
and
for
transition
condition,
rise
to the
is entirely
upper
surface. drag
while
point
is
preserving
bubble
has
been
corresponds point
suction
formation
to
precedes peak
again
of a leading-edge
therefore,
analogous
upstream
It is particularly
the
of the
to
a direct
lower-surface
transition
concerned,
to be
the
is shrinking
scaling
upper little
distribution
of n along
shows
the
contributes
is believed
pressure
bubble
this
distribution
case
where
of a 16%c long bubble.
upstream
before
bucket,
surface
distribution
of attack,
Beyond
travels
lower
adverse
upper-surface
angle
pressure
quite
predicted
on the
of the
in this
in spite
higher
effects
6-3,
At
6-5(b),
summary
drag the
separation
used.
the
by the
that
and
in Fig.
bubbles
top
length
indicates
separation
to laminar
bubble.
how
a slightly
and
effect
analysis
point
to the
transition
viscous
This
are
separation
in Fig.
to disappear
shown
to observe
seen
bubbles
is evident.
is clearly
the
be
The
of laminar
corresponds
short
surface
of n was
is shown
of the upper-surface
identified.
the
This
6-6(b),
interesting its
The
lower
as can
value
of the destabilizing
of separation. In Fig.
same
bubble
finally,
and
is highest.
upstream
is about
drag.
consequence
only
the
of the
upper In fact,
development
part
Fig.
the
bubble
to be amplified
boundary-layer
the
Both
it appears
to a rise
that
in Reynolds
number.
Eppler This on model
airfoil gliders.
was
designed
It also was
more recently
E387 than tested
Airfoil
twenty in the
years
ago
NASA
and Langley
is intended
for
Low-Turbulence
use
109
0
I
o_
0
m
o=
2_ c
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L
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•
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110
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F_
ii ,,
s_ s_
= SF
sin-l(1/A1
upon
until
(A.29)
e -300(h7/c)
the
(A.30)
undershoot
merges
with
the
inviscid
velocity
distribution.
The equations
Governing
above in the
shape
factor
inverse
distribution
is used
to drive
the integral
boundary-layer
mode.
equations:
-
d62 _ ds
(cs12)H32 - Co
[_ 3(cy/2)H32 [ H12+2
dC,C_- _
x -
4.2(C_,
+hdH_] u as j _2H32(H12- 1)
+CD--_52
dH32 ] H12 + 2 ds J H32(H12 -1)
(A.31) (A.32)
x -
C¢ )
(A.33)
148 where 3
0.015H32 --
6 = 62 Closure
(A.34)
H12
3.15 + H12 1.72-
1)
(A.35)
+ 61
correlations: CD = f x [CD]D,-el.
(A.36)
m
where /
\l_ f = { 1
+ 1 _-
r = 15 -
0 < _-=_- < 1
],
., -_(_-qa-_--1)
(CD,.n.,, (CDm..
(A.37) --v__: • > 1 t2
1000 hT"
(A.3S)
E
(A.39) [CD]O_a = ciU_. p + 2C_(1 - U_.p) _
/_2
(A.40) (A.41)
4
H,_ =
1 • rg_2-B.2o
*L
r.r 22120
cf>O
1 r_v '
°, J
-- r Ha2-H32_ l -I-
(A.42)
A_
Ht2o [
c2
¢8
J
cf
