Transition caused by the Laminar flow separation

MISSING IMAGE

Material Information

Title:
Transition caused by the Laminar flow separation
Series Title:
NACA TM ;
Physical Description:
14 p. : ill. ; 27 cm.
Language:
English
Creator:
Maekawa, T
Atsumi, S
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington
Publication Date:

Subjects

Subjects / Keywords:
Aeronautics   ( lcsh )
Laminar flow   ( lcsh )
Boundary layer   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
An experimental investigation of the effects of the geometry of body surface, Reynolds number, stream turbulence, and a roughness element (wire) on the reattachment of separated laminar boundary-layer flow on a bent flat plate is presented and discussed. The flow mechanisms determining reattachment of the boundary layer are analyzed and discussed.
Bibliography:
Includes bibliographical references (p. 10).
Funding:
Sponsored by National Advisory Committe for Aeronautics.
Statement of Responsibility:
by T. Maekawa and S. Atsumi.
General Note:
"Translation of "Sōryu-Hakuri ni tomonau Sen̕ i ni kansuru Kenkyū." Journal of the Society of Applied Mechanics, Japan, vol. 1, no. 6, Nov. 1948."
General Note:
"Report date September 1952."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778661
oclc - 44794299
sobekcm - AA00006194_00001
System ID:
AA00006194:00001

Full Text
S4-7-m -







q. f i "


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1352


TRANSITION CAUSED BY TBE LAMINAR FLOW SEPARATION*

By T. Maekawa and S. Atsumi


INTRODUCTION


A great number of discussions have long been made to explain the
cause of transition from laminar to turbulent flow in the boundary layer;
none of them has as yet led to a definite conclusion. Many attempts have
been put forward to calculate the transition theoretically by considering
the stability of the boundary layer with superimposed infinitesimal dis-
turbances. But the results do not agree with experiments except in the
case of the transition in the flow between concentric rotating cylinders
discussed by Taylor (ref. 1). (According to Nature, December 14, 1946,
it is reported that Dryden has succeeded in showing the agreement between
the calculated result by stability theory and the transition observed in
a wind tunnel of low stream turbulence, but we know nothing about the
details.)

Another example of the breakdown of laminar flow is the case when
the boundary layer flow separates from the surface while it is laminar.
There are two cases; in one case the separated flow leaves the surface
forever, while in the other case the separated flow becomes turbulent
due to its instability and reattaches to the surface as a turbulent
boundary layer. The latter case is a kind of transition, and there are
many experimental examples in which such a sort of transition appears.
The transition which occurs when there is a steep rise in surface pres-
sure distribution seems to be mostly due to this mechanism. The sudden
change in drag or maximum lift occurring in a certain range of Reynolds
numbers is considered to depend upon whether or not the separated flow
reattaches to the surface. In spite of such an importance in practice,
very few studies have been made concerning this problem, except for
Von Doenhoff's kinematical speculation deduced from a simple experiment
(ref. 2), Tani's postulate of constant Reynolds number at the laminar
separation point based on dimensional analysis (ref. 3), and Inoue's
discussion of Von Doenhoff's speculation (ref. 4). Experiments were
therefore made to examine the effects of the geometry of body surface,
Reynolds number, stream turbulence and a roughness element (wire) on the
reattachment of separated flow, with a view to inquiring about the
mechanism of transition caused by separation and hence to understanding
the effect of Reynolds number on the flow characteristics.

"S5ryu-Hakuri ni tomonau Sen'i ni kansuru Kenkyu." Journal of
the Society of Applied Mechanics, Japan, vol. 1, no. 6, Nov. 1943,
pp. 187-192.







2 NACA TM 1352


I. MEASURING APPARATUS


The wind tunnel used for the experiments was of the N.P.L. type,
having a test section 50 cm x 50 cm. The upper wall of the tunnel could
be opened by means of hinges so that we were able to change as well as
work on the model. The upper wall had a central groove, into which a
sliding cover was put so as to be able to move without air leakage in
the stream direction. The cover carried a measuring instrument, pitot
tube or hot-wire anemometer, and the micrometer which moved the instru-
ment up and down. One of the side walls was made of plate glass which
enabled us to observe the flow from outside.

The models used were a wooden hollow flat plate 5 cm thick, 60 cm
long, 50 cm wide, and a duralumin flat plate 7 mm thick, 70 cm long,
50 cm wide, each spanning the test section of the wind tunnel. The
rear part of the model, 30 cm long, was hinged to the forward part, so
that the model could be bent at the hinge point. Since a sharp rise
in pressure occurs at the bent corner, the separation of flow was fixed
there. Thus, we could simplify the experimental conditions by elimi-
nating the parameter concerning the point of separation. In order to
vary the condition at the separation point, we varied the angles a
and p of the forward and rear parts respectively with respect to the
direction of wind tunnel stream (fig. la).

The separated flow was examined by (1) measuring the turbulence by
the hot-wire anemometer, (2) observing the flow pattern as indicated by
the lampblack in oil put on the surface, and (3) measuring the dynamic
pressure 0.15 mm above the surface by a pitot tube in contact with the
surface. The hot-wire was a platinum wire of 0.05 mm diameter, and had
a time constant M = 4.2 X 10-3 sec. It was used in connection with a
compensating circuit. The pitot tube was made from copper tubing of
0.5 mm outside diameter, flattened to a mouth of 0.1 mm inner diameter
and 0.1 mm thickness. The pressure distribution along the surface was
measured by a static-pressure tube made from copper tubing of 2 mm out-
side diameter, the end of which was closed and in which 4 holes of
0.2 mm diameter were drilled around the tube.


II. GENERAL OBSERVATION BY THREE METHODS OF MEASUREMENTS

(A) Comparison Between the Pitot Tube

and Lampblack Measurements


The measurements were made on a duralumin plate with a = -9030'
and p = 70 for two wind speeds, 8.1 m/sec and 15.3 m/sec.







NACA TM 1352


The dynamic pressure measured by the pitot tube in contact with
the surface is shown in figure Id. The dynamic pressure is zero at
the bent corner, for example, the separation point, becomes negative,
then increases suddenly, reaches a maximum, and finally decreases
gradually. The lampblack method enables us to observe, as shown in
figure If, the region of lampblack at rest, the region of lampblack
accumulation, and the starting line of reverse flow. When the Reynolds
number increases, the region of lampblack at rest disappears, and, at
the same time, both the region of lampblack accumulation and the starting
line of reverse flow move upstream. Generally speaking, the downstream
edge of lampblack accumulation coincides with the position where the
dynamic pressure begins to increase, and the starting line of reverse
flow coincides with the maximum point of the dynamic pressure. It
seems therefore that a thin elongated vortex extends downstream of the
separation point when the separated flow reattaches to the surface as
a turbulent boundary layer.

The pressure distribution along the surface is shown in figure le.
As the Reynolds number increases, the extent of the separation region
decreases accompanied by a considerable change in the pressure distri-
bution; thus the pressure coefficient at the bent corner is almost
doubled and the resistance halved. It will be seen therefore that there
is a considerable effect of Reynolds number due to the change of the
extent of the transient region, although the position of separation is
fixed. We expect to consider this problem in more detail at another
time.

There are cases where the dynamic pressure indicated by the pitot
tube does not decrease but sometimes increases gradually after reaching
a maximum value. In such cases, the reattachment of the separated flow
is found to be incomplete, now attaching and now leaving the surface
in turn.


(B) Observation of Turbulence by a Hot-Wire Anemometer

The measurements were made on a wooden plate with a = -5o and
p = 130 for wind speeds Us at the separation point from 14.6 m/sec to
16 m/sec. Moving the hot-wire downstream keeping the distance at about
0.5 mm from the surface, we observed just downstream of the separation
point velocity fluctuations of low frequency, the amplitude increasing
but gradually. At the same time, high frequency fluctuations occurred,
the amplitude being still small. Finally, violent high frequency fluc-
tuations appeared. Downstream of the point of reattachment, the fre-
quency became lower and the amplitude decreased.

When the hot-wire was moved away from the surface, the fluctuations
became smaller and smaller, until they were equal to those of the general
stream. The boundary of this region looks similar to the upper boundary
of turbulence diffusion as speculated by Von Doenhoff (fig. Ib).







NACA TM 1352


The description mentioned previously is quite similar to that
hitherto observed in the usual transition of the boundary layer. We
are therefore led to the suggestion that most of the transition occurs
by means of such a mechanism. According to Von Doenhoff's speculation,
the triangular region ABC of figure lb is considered to be quiet, while
according to our observations (A) and (B), a vortex seems to exist here,
thus causing violent velocity fluctuations. We will consider this point
again in section VI.

When the flow leaves the surface without reattachment, the circum-
stances are different from those described previously, the velocity
fluctuations decreasing as we approach the surface and increasing as we
proceed downstream.


III. VARIATION OF THE REATTACHMENT WITH a AND B


Measurements were made on a duralumin plate by the pitot tube
method to determine the variation of the point of reattachment for
various values of a, I, and wind speed. The wind speed Us, the
speed outside the boundary layer at the separation point, was varied
from 7 m/sec to 15 m/sec. The results are summarized in table 1.

LUs
If we construct a Reynolds number from Us and L, RL = Us
v
where L is the distance between the bent corner and the point of
reattachment, we find that the values are approximately constant inde-
pendent of the wind speed for a fixed set of values of a and 3.
Plotting the values of RL on the corresponding geometrical configura-
tions, we find the points lying approximately on a straight line for a
fixed value of a.

This result reminds us of Von Doenhoff's hypothesis, but it is dif-
ficult in our case to define the direction of separated flow, because
there is no tangent line at the bent corner. Most naturally, we first
assume the direction to be the bisector of the angle formed by the down-
stream surface and the elongation of the upstream surface. This bisec-
tor and the straight line through the reattachment points makes an
angle 7 equal to 13040' independent of the value of a, but the dis-
tance from the separation point to the intersection of the two lines
gives the Reynolds numbers Rx = 17,200, 13,300, and 12,300 for
a = -9030', -12030', and -1030', respectively.

We then assume that the direction of separated flow coincides with
that of the general stream for the case of a = -9030', and that it
increases by an amount .a, where la = -930' a. The distances Rx
rfj







NACA TM 1352


and angles 7 defined similarly as in the foregoing paragraph then
become Rx = 26,000, 25,000, 23,700 and ? = 17030', 17030', 15050'
for a = -9030', -12030', -14030', respectively. Averaging the results,
we have Rx = 25,000 and 7 = 170 (fig. 2a).

If the angle 0 is increased such that the Reynolds number RL
exceeds 75,000, the reattachment becomes incomplete and, as already
mentioned, the dynamic pressure does not decrease after reaching a max-
imum. Such values of RL, which are marked by parentheses in table 1,
do not lie on the straight line. Even when 2 is below the limiting
value, but close to it, a similar tendency appears -jhen the wind speed
is low.

The velocity profile of the boundary layer at the separation point
was measured for the case a = -9030', 2 = 70. The values of Reynolds
number Re based on the momentum thickness were found to be from 178
to 249 when the separated flow reattaches to the surface completely.
So Tani's postulate on the critical value of Re appears to ha.e to
be reconsidered in our case.


IV. EFFECT OF STREAM TURBULENCE


In order to see the effect of stream turbulence on the reattachment
of laminar separation, measurements were also made with a square mesh
grid (composed of wooden round bars of 6.35 mm diameter at a mesh
length 25.4 mml placed at a distance 63.4 cm upstream, or with a wire
screen (composed of 1 mm diameter wire at a mesh length 5 mm) placed at
68 cm upstream of the leading edge of the model. The values of turbu-
lence intensity at the separation point were 1.7 percent'and 1.0 per-
cent, respectively, while the value for the bare tunnel was 0.15 percent.
The results are also summarized in table 1.

Analyzing the results by the first method, in which the direction
of separated flow is assumed to be the bisector of the angle, we have
Rx = 13,800, ? = 150 for the case of wire screen, and Rx = 17,200,
S= 15040' for the case of grid. Analyzing by the second method, in
which the direction of separation is assumed to coincide with the
general stream for the case a = -9030', and to increase by an amount
i I for the other cases, we have Rx = 25,000, 7 = 20030' and
Rx = 26,000, ) = 22050' for the cases of wire screen and grid, res-
pectively. It seems not unreasonable to average the results to get
Rx = 25,000 and 7 = 21050'. This supports the second method of anal-
ysis in that the final result becomes simpler. The values are shown in
figure 2b.







NACA TM 1352


Similarly to the previous result, when the value of RL


exceeds


a certain critical value, the reattachment becomes incomplete, the
dynamic pressure showing no decrease after reaching a maximum, and the
value of RL deviates from the straight line. The critical value
of RL seems to lie in the range from 48,000 to 56,000 both for the
cases of turbulence intensity 1.0 percent and 1.7 percent, although the
measurements were insufficient to determine the exact value.


V. THE EFFECT OF A PIANO WIRE PLACED ABOVE

THE SEPARATION POINT


Measurements were made to examine the effect of a piano
0.5 mm diameter, which was placed at various distances, y =
4.09, 5.0, and 7.0 mm, above the bent corner. The angle P
from 60 to 90 and the wind speed from 8 m/sec to 16 m/sec.


wire of
2.04, 3.0,
was varied


Two kinds of Reynolds number may be considered as the parameter
representing the effect of the wire. The one is the Reynolds number Rd
based on the diameter d of the wire and the wind speed U at that
position, and the other is the Reynolds number Ry based on the
height y of the wire above the surface and the wind speed U. Since
the thickness of the boundary layer was from 2.2 mm to 2.4 mm, the value
of U must be smaller than Us when y = 2.04 mm. When y is large,
the value of U is again expected to become smaller than Us due to
the character of potential flow. However, we simply adopt Us in place
of U, because we made no measurement.

The relation between RL and Ry is shown in figure 3 for dif-
ferent values of 0. The measured points lie nearly on a single curve,
there being a tendency, however, such that RL is larger when Rd is
larger. This tendency seems to be associated with the fact that the
drag coefficient of the circular cylinder decreases from 1.27 to 1.13
in the range of Rd from 250 to 530, which correspond to the present
measurement. Determining the faired values of RL from the curves of
figure 3 for various values of Ry and 0 (table 2), and plotting RL


on the corresponding configurations as shown in
lines drawn through the points, except for 0 =
to a point which is R, = 25,000 distance from


figure 2c, the straight
8, seem to converge
the corner in the direc-


tion of the general stream rotated by an angle 1 'a = 1030'. The values
2


of angle 7
in table 2.


made by each straight line with the reference line are given
It seems somewhat strange that Rx remains constant even







NACA TM 1352


if y exceeds 900. But it is to be noted that the experimental results
cannot be explained if we assume the turbulence to diffuse right from
the position where the wire is placed.


VI. DISCUSSION OF RESULTS AND CONCLUSION


We find from the preceding results that the reattachment of the
separated flow may be simply interpreted in terms of Rx and r as
put forward by Von Doenhoff, and that a reverse-flow vortex exists and
hence the velocity fluctuation is most violent in the region which is
considered to be quiet according to Von Doenhoff's hypothesis. These
two facts apparently contradict each other, but they may be reconciled
if we consider that Von Doenhoff's hypothesis is responsible for
explaining the transient process from the initial separation of lami-
nar flow to the finally arrived stationary state in which the flow
reattaches to the surface, and that the reverse-flow vortex plays an
important role in the stationary state. In other words, the separated
flow becomes unstable and produces turbulence, which extends to the
surface and assists the separated flow in reattaching to the surface.
During this process, the fluid in the triangular region remains at
rest, but it develops into a vortex due to the dragging action of the
outside flow. The vortex thep acts positively to attract the separated
flow toward the surface. Since the vortex maintains itself by the cir-
culation continuously supplied from the separation of the boundary layer,
there must be a balance between the strength of the vortex and the cir-
culation at the separation point. If the Reynolds number RL becomes
too large, which is proportional to the length of the vortex, it becomes
impossible to supply circulation sufficient to maintain the vortex
against the dissipation. Then the vortex becomes unstable and makes
the reattachment incomplete. This explains the existence of the upper
limit of the value of RL. It seems at first difficult to understand
the fact that Rx is independent of the stream turbulence, but such a
circumstance may be suggested from the researches made by Schiller and
his coworkers (ref. 5), which are briefly as follows: For flow in pipes
and channels, there is a lowest critical Reynolds number such that for
Reynolds numbers less than the critical value all disturbances, however
great, are damped out sufficiently downstream. The critical value
is 1160, when the radius a of the pipe, the half width a of the
channel, or the thickness 5 of the boundary layer is used as a refer-
ence length in defining the Reynolds number. Schiller suggests that a
vortex street arising from the edge of the entry develops into turbulence
when Z/v exceeds 1170, where Z is the circulation of the vortices
contained in the length equal to a or 5. It will be seen therefore
that the lowest value of the Reynolds number of the vortex street coin-
cides roughly with the lowest critical Reynolds number of the flow, and







8 NACA TM 1352


that the vortex street always develops if Z/v > 1170 and 5U/v > 1160,
but dies away if one of the inequalities does not hold. This fact seems
to be essentially the same as our finding that R, is constant. It will
be most natural in our case to take Z equal to ulb, where ul is the
velocity outside the boundary layer and 5 is the thickness of the
boundary layer; Z/v then becomes the Reynolds number of the boundary
layer. If Z/v is less than 1170, then the separated flow is stable
so that no vortex is produced, and therefore insufficient circulation is
supplied for maintaining the reverse-flow vortex even if it were pro-
duced. This is considered to lead to the incomplete reattachment of
the separated flow. We observed several examples in which the dynamic
pressure indicated by the pitot tube in contact with the surface is
constant after reaching a maximum value, when RL is comparatively high
and the wind speed is low. Moreover, according to the additional meas-
urements for the case a = -9030', D = 110, which is very near to the
critical value of RL, the flow was found to be stable for Us higher
than 10.35 m/sec, but unstable for the range from 3 m/sec to 3 m/sec.
The Reynolds numbers 8Us/v corresponding to the unstable cases are
from 1326 to 381, thus agreeing with the fact mentioned previously.

The criterion due to Tani is formally the same, but the value given
by him is too high. Moreover, this is the necessary condition, but not
the sufficient condition, so that the condition alone cannot determine
what happens.

Summarizing the results of our study, we have as follows:

1. Von Doenhoff's speculation is valid for explaining the transient
process during which the initial separation of laminar flow develops
into a stationary state with the reattachment of flow. But in the sta-
tionary state, a reverse-flow vortex produced downstream of the separa-
tion point attracts the separated flow toward the surface.

2. The stability of the separated laminar flow breaks down when
the Reynolds number Rx exceeds 25,000, this critical value being inde-
pendent of the stream turbulence. But the instability does not appear
when the Reynolds number based on the boundary layer thickness at separa-
tion is less than 1200.

3. After the breakdown of laminar flow, the turbulence spreads at
a certain angle which increases as the stream turbulence increases.

4. The reattachment of the separated flow becomes incomplete when
the Reynolds number RL exceeds a certain critical value, which seems
to decrease as the stream turbulence increases.







NACA TM 1352


5. Even if the separation point is fixed, the extent of the separa-
tion region affects considerably the pressure distribution as a whole as
well as the characteristics of the reattached boundary layer. Therefore,
the experimental data on a turbulent boundary layer whose condition of
transition is not clear seem to be insignificant.


VII. NOTE ON TRANSITION FOR SPHERES AND CIRCULAR CYLINDERS


According to Von Doenhoff, the critical Reynolds number Rc where
a sudden decrease in drag of spheres or circular cylinders begins is
given by


UO 2
Rc = R U tan
X Us tan -
2


where UO is the speed of undisturbed stream. Rx is independent of the
stream turbulence as mentioned previously. UO/Us is also nearly inde-
pendent of the stream turbulence. Hence, assuming Rc to be propor-
tional to the conventional critical Reynolds number of the sphere, we
get for the so-called turbulence factor (T.F.) the expression


Rc0 tan Z
T.F. co = -
Rc tanO -


where the subscript 0 refers to the condition of the free atmosphere.
Assuming the relation between the turbulence factor and the percentage
intensity of turbulence in the form T.F. 1 = 84(u'/U), and adjusting
the constant so as to make 7 = 170 for u',U = 0.15 percent, we have


7 = 15.10 x T.F. = 15.10 + 12670 -
U


If we estimate j by this formula for the case u'/U = 1.0 and 1.7 per-
cent, we have 7 = 27.80 and 36.70, respectively, which, however, are
far greater than the experimental values mentioned above. This suggests
the importance of taking account of the effect of surface curvature







NACA TM 1352


distribution. It seems therefore necessary to make extensive measure-
ments on the effect of surface curvature distribution, before we are
in a position to discuss the effect of Reynolds number and, especially,
the effect of stream turbulence (concept of effective Reynolds number)
on the characteristics of flow.


Translation by Itiro Tani
University of Tokyo
Tokyo, Japan




REFERENCES


1. Taylor, G. I.: Phil. Trans. Roy. Soc., A, vol. 223, 1923, p. 239.

2. Von Doenhoff, A. E.: Proc. Fifth Int. Congr. App. Mech., 1938,
p. 311.

3. Tani, I.: J. Soc. Aero. Sci. Japan, vol. 7, 1940, p. 229.

4. Inoue, E.: J. Soc. Aero. Sci. Japan, vol. 10, 1943, p. 107.

5. Schiller, L.: Proc. Fifth Int. Congr. App. Mech., 1933, p. 315.







NACA TM 1352


TABLE 1


TABLE 2


Ry 1500 2000 2500 3000 4000 5000 6000 7000

3 = 60 2.20 2.58 2.92 3.19 3.60 3.85 4.o4 4.20
4 70 2.15 2.60 3.00 3.30 3.76 4.10 4.34 4.5o
RL x 10 g8 2.01 2.46 2.90 3.29 3.93 4.41 4.70 4.81
90 2.12 2.69 3.23 3.70 4.51 5.10 5.44 5.50

7 138.5 73 43.50 31 24.5 24.5 21.50 200 190







NACA TM 1352


la
Bent plate in
wind tunnel


0A-_
i
e~r


lb
Van Doenhotf's
speculation


1c
reverse -flow
vortex



id
Pitot tube
measurement




le
surface pressure
distribution




Region
if of
lampblack measurement


Quiet region
Region of turbulent diffusion
, x Direction of
separated flow


Figure 1.








NACA TM 1352 13








a a
N ON

oob u qj/ *a

a x 0 0 *0



CA r o
I'CL
U- a + 0 /-
.. / o_ ,


(z/
1 0
E" _j

0 0o






< O O
S-0
I- It














"- -
0.












O CL
I








0 a ao* 0
0 o 0 0 u,
0 E O -

0q1 0al ow
I ,,111 :ZI c r c
,,/ .. ,,
a n-,".
"u -r 7 ..1







NACA TM 1352


RL
1K)04


Figure 3.- Effect of a piano wire on the position of reattachment of
separated flo'.v.


--- ... i- 6-1-t2 innf









1 cG




0 to 00
3 0. qj 1

r
E LI o jm*

o co -y


r-

gI


.0
E-


0 2n z m
- CO s- *? > V
",^ ""
0S < a |
z to M'1
oj ~ CJ "^ j^ !
S ~ia^u aS


zru


Wh
' *0
1. O>
; aa


pD





So
I"



U:.d
.0
.5
*ti





V
.01
B a
I!
u "


ti
^1
B
dc
n


v.i w C .o .
-4 -0 -4 0



0_ (D
'0;

> S .0 ,
a hg m -m~cd
0 l o 0e (
i t: Illi li il 7
I I IC 0- L
0 0 :1 cd U > 1/'
r o u 0 w 2* L 24~ z 8 9.


m

Z

*S a- VJS f
04. 0. M 4
a0 g

o iS
jls^-sat








40Z a m
SU 0L Fl
z^ :4


0> ^ 1 C4 S




ti gz a SS aPt
C4 ci
EWJSM


cc 00
o o0 D o .-
O !:% 0- S-

Cd csz M
C~;~p O2C

.00
cda c.a w 'D


o 0 L M Iu > v


:i 0i 2 4a ,4E


I0

A. ')OL


.$ fz4 U2 0z ~


D4 ;,
VLUO
kS CDco





'Sca 3Sa
cu a; 5 S"







223 0 U > mi c
0 0
mcl1 I'


"y a 5. vjino
SAti 0 4
Sf i *^*
rr .
g: 8S..3.



%

'V.-
V0

az m
* CO Vt '


aE a




ro.0
0fl 5 cd
i o :..








: .0 =
0t
" '0
.Yas

5 "a "
rI
O

0a.0
ia^5


-M -S4 -Q

in cm = w
E4 .i CL -


1 e 0 nii
z s i .S
t o o I '0<

'_; F im
E <
V4 2mZ4 wo

4 i 10; ~ i~


I










En 0 *-r O0..
- L 'C 4
*' a; M .- _C 0

c. GoC -4 -3 co
8 .5 = ^ I
0 0 :3. C l 0 '
o o e u 9 s
o C o [n X W Go


- a
0 -o
N "
il 0

0
I .. S


C 4

CU u
- 0
*S.I
0



.0 0
u

- U


0
0
a
In




4 i

zzI
1: L


o o t So m gp L

E- < -


0 *^ Jc |- 0


0 0 w 01 44 > m
CU. 4
~~4a 02 0 *.1


0

0
a
0
u


0"2
in
co E
,I


U-
zz


* 5



' >a

I? g
Co


0


*<
'0
0
0 U
S.3
0' 0
I I.
- 0

0Q 0
0 0
.5 0


Ca P
=' 8'^ S^" -.^ ^ p
wl ga4 .34US o T 0 2r
lp I o ; IS
id Ea r L b Wo ~ ooa~

CI I 00' V c c
-Z) C4r
0~ 0 E- 0
E-4 1. 0~P E-4* E
Z ca uii 0 -Z E Sin 0
0~ V .C 0 01
I I 11 ; I I CU 0
v0 -
0 cd >,: cd :
rk 5: me U w x z oo

C..' 02 .4 :i d2 '


^^g^ .r S *
CU l 0 *-gi ||i 0

ca Z 0 ; *S s'.s
v "de o.

P1 5 'a s Q

oU C i lcgl l

2 z I t JS en
;Z1

CD ( ci r. o 7
~.9 o- z: I
--


0 o2
F-~ m 0~ E-0-Lj,
0a.0 *m 4
:CU Ot.CUO.O i


5 c
'I ?
1 I
N






0
a.
a S
a 0
S..
IU -
- 0
0
C


0s 0
c s
3 0
0
. 0










- cw g *i

- C 1C


. s l li
-U CW E




o a m d
- C cI= G, CU0"
O O uwww'Q ~ n>


to w
L. w-
W CT .t .^. Co
ri~ ~ ~ U1?t 'ci S4oC



d~Cd
*4 -4 0I
S-< CI'- ..D < B^l < S B fl.




s li l
o 0 0m~o 4ZN

k E auos

.1 CU


Be I
0.





-, > -

C U
4 CO









< A2 cis^
Z* -tBie- Q.


* ,3 ai z *
W C-



IU zI -;
. (I A

0' 0 CT 4W MaS ,










UNIVERSITY OF FLORIDA

S1262 081058199