Determination of Rate, Area, and Distribution of Impingement of Waterdrops on Various Airfoils from Trajectories Obtained on the Differential Analyzer
RM No. 9A05
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RESEARCH MEMORANDUM
DETERMINATION O F RATE, AREA, AND DISTFUBUTION O F IMPINGEMENT
O F TJ\rATERDROPS ON VARIOUS AIRFOILS FROM TRAJECTORIES
OBTAINED ON THE DIFFERENTIAL ANALYZER
BY
A. G. Guibert, E. Janssen, and W. M. Robbins
University of California
N 66L80972
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(NASA CR O R T M X O R AD N U M B E R )
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(CATEGORY)
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AiR~ j n ,-NATIONAL ADVISORY; COMMITTEE
FOR AERONAUTICS
WASHINGTON
February 16,1949
I
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NATIONAL ADVISORY COMMTTTEX FOR AERONAUTICS
RESEARCH r4EMORANIxTM
DEFERMINATION QF RATE, AREA, AND DISTRIBUTION OF IMPINGEMENT
OF WATWlwOPS ON VARIOUS AIRFOIIS FROM TRAJECTORIES
OBTAINED ON W DIFFWENTIAL ANALYZER
By A . G. Guibert, E. Janeeen, and W. M. Robbina
SUMMARY
The t r a j e c t o r i e s of waterdrops i n a i r flowing over a i r f o i l s a r e
determined f o r three a i r f o i l - angle-of-attack combinations using the
d i f f e r e n t i a l analyzer t o solve t h e d i f f e r e n t i a l equations of motion
of the waterdrops. From these t r a j e c t o r i e s the r a t e of water impingement, t h e area of impingement, and the d i s t r i b u t i o n of impingement
a r e determined as functions of two dimensionless moduli.
Comparisons a r e made of the r a t e of water impingement on these
a i r f o i l s and the r a t e of water impingement on cylinders.
INTRODUCTION
If a body of any shape and s i z e moves through a cloud, some of the
waterdrops i n i t s path w i l l tend t o impinge on the surface of t h a t body
over an a r e a which w i l l vary according t o the s i z e of the drops, the
speed of t h e body, and s o f o r t h . Other drops o r i g i n a l l y i n i t s path
w i l l be c a r r i e d around the body and w i l l n o t impinge- Studies have
been made of the r a t e and d i s t r i b u t i o n of impingement of waterdrops
on cylinders and two d i f f e r e n t a i r f o i l s by means of numerical i n t e g r a t i o n of the d i f f e r e n t i a l equations of the motion of the drops
(references 1, 2, and 3 ) and on cylinders, spheres, and ribbons by
s o l u t i o n of these equations on a d i f f e r e n t i a l analyzer (reference 4 ) .
References 1 and 2 both made the assumption t h a t the velocity and
s i z e of the drops were such t h a t Stokes' l a w of r e s i s t a n c e w a s followed.
References 3 and 4 did not make t h i s assumption, which i s not applicable
a t t h e v e l o c i t i e s of airplanes and f o r the drop s i z e s prevalent i n
clouds. These references employed instead the experimentally determined
drag c o e f f i c i e n t f o r spheres as a b e t t e r approximation t o the drag
c o e f f i c i e n t of the drops.
NACA RM No. 9AOg
2
I n the praoent study, the r a t e and d i s t r i b u t i o n of impingement of
waterdrops on a symmetrical, 15-percent-thick, Joukowski a i r f o i l a t
angles of a t t a c k of 00 (Case I ) and z0 (Case 11) and on a cambered
(a = 1 mean l i n e ) , l?-percent-thick, Joukowski a i r f o i l a t 00 angle
of a t t a c k (Case IV) are determined Using a d i f f e r e n t i a l analyze; f o r
s o l u t i o n of the d i f f e r e n t i a l equations and employing the experimentally
determined drag c o e f f i c i e n t of spheres t o approximate the drag
c o e f f i c i e n t of the waterdrops. Figure 1 i s a drawing of t h e three
a i r f o i l cases (Case I11 wa8 t o have been a study of the symmstrical,
l?-percent-thick, Joukowski a i r f o i l a t 4' a n g l e of a t t a c k , b u t i t
w a s decided t o study Case I
T i n preference t h e r e t o . )
This p r o j e c t w a s under the general d i r e c t i o n of L. M. K. Boelter.
The authors wish t o acknowledge t h e advice of John W. Hazen i n the
d i r a c t i o n and implementstion of the research program and the a s s i s t a n c e
of R . Peck and M. P o t t e r i n making the necessary conputati'ons f o r
presentation of the data and a l s o the a s s i s t a n c e of the operators of
the d i f f e r a n t i a l analyzer under E. Janssen and G. N. B r i t t l e .
This work w a s conducted under the sponsorahip and with the f i n a n c i a l
a s s i s t a n c e of the National Advisory Committee f o r Aeronautics.
SYMBOLS
a
acceleration of rtrop, f t / s e c *
A
projected area of waterdrop, f t *
C
chord length of a i r f o i l , f t
CD
drag c o e f f i c i e n t of drop, [1J*
E
EM
percentage catch
t o t a l percentage
f
drag force, #
M
r a t e of impingement of waterdrops on a body, l b / h r f t span
P
velocity of drop r e l a t i v e t o air, f t / s e c
r
radius of drop, f t
*
Dimensionloss
The abbrsviation, l b , represents pound mass;
t h e symbol, #, represents pound force.
-K-x-
based on maximum
113
thickness of a i r f o i l [ 13
c
*P
Reynolds Modulus f o r drop a t r e l a t i v e v e l o c i t y P, [l]
Rv
Reynolda Modulue f o r drop at free-stream velocity, [l]
8
p o s i t i o n of Impingement on eurface of a i r f o i l , measured fron
chord line, divided by chord length, [l]
S
furthest p o s i t i o n of impiwement on surface of a i r f o i l (lee.-,
trajectory t
e n t ) , measured from chord l i n e , divided by
chord length>l]
t
time, sec
ua
v e l o c i t y component of air parallel t o chord line, f t / a e c
Ud
v e l o c i t y component of drop parallel t o chord l i n e , f t / s e c
U
free-stream velocity, f t / s e c
va
v e l o c i t y component of air normal t o chord l i n e , f t / s e c
Vd
v e l o c i t y component of drop normal t o chord l i n e , f t / s e c
W
l i q u i d water content of cloud, l b / f t 3
X
distance from the a x i s normal t o chord l i n e which i n t e r s e c t s
leading edge a t chord l i n e i . e . , distance f r o n y-axis),
divided by chord length,
d?
,
U’
u
- = #, [l]
b d
dT
distance from the axis p a r a l l e l t o chord l i n e which i n t e r s e c t s
leading edge a t chord l i n e ( i . e . , diatance from x-axis),
divided by chord length, [l]
YO
distance of a t r a j e c t o r y from the x-axis a t x = -a, divided
by chord length, [l]
Ya
fd
Ya
mass density of a i r , l b / f d
.
4
7d
NACA RM NO. 9A05
mass density of drop, l b / f t 3
P
absolute v i s c o s i t y of a i r , Ib/sec f t
$
Scale Modulus, 9 C A , [1]
7
U
Time Scale, t- [l]
C'
e
angle of r e l a t i v e velocity vector from x-axis [1]
U
angle of a t t a c k of a i r f o i l , deg
Vd
Subs c r ip t a :
A
airfoil
C
cylinaer
L
lower
U
upper
1
f i r s t impinging t r a j e c t o r y
2
second impinging t r a j e c t o r y
ANALYSIS
I n a cloud, the motion of a waterdrop which results when a body
moves through t h a t cloud with f i n i t e v e l o c i t y i s caused by t h e drag
created by f l o w of tha displaced a i r r e l a t i v e t o the waterdrop. This
r e s u l t i n g motion i s the same as if t h e waterdrop had bsen suspended
i n a i r flowing over tha s t a t i o n a r y body with the same v e l o c i t y .
Making a force balance on the waterdrop (see f i g . 2) one obtains:
' I I F = O = m - f
ZF,
= 0 = m
-- f
dud
at
COS
8
at - f sin e
E F Y = o = m dvd
whera
f , t h e drag force, is:
NACA RM me. 9AQ5
i
5
c
and
COS
e = - ('a
- Ua) - U a - Ud
P
-
P
*
Multiply both sides by
L
at
b d
C 2 p- 2
-r Y-a
u 3 2 r IJ Y f l
S u b s t i t u t i n g i n equations ( 5 ) and (6) the relatioriships:
6
give s
Equations (7) and (8) are the desired equations for the twodimensional motion of a waterdrop in an air stream flowing over a b o a
For solution of the differential analyzer, these equations must be
m g e d as follows:
r
J
1
NACA i3M No. YAO3
n
I
n
-J
d(P/U)2
2(P/U)
(Inverse integrator)
Knowledge of the magnitude of the quantities CDEP ( t h e ratio of the
24
actual drag coefficient to the drag coefficient given by Stokes' law) and
the velocity components of the air stream, ka and fa, as a function of
the location of the waterdrop relative to the body, must be available for
the solution to proceed on the differential analyzer. The variation
of
c
Bwas taken from table I in reference 4.
24
Plots of ia and
ia,
the velocity components of the air stream, as functions of position
relative to the alrfoils under consideration were supplied by Ames
Aeronautical Iaboratory, Moffett Field, California.
Finally, having fixed .Jr, the Scale Modulus (presented in reference 5 ) ,
and Ru, the Reynolds Modulus of t h e drop based on free-stream velocity,
solution of the equations c a n begin provided initial conditions for a
trajectory are known. If it were possible to start the tradectory at
infinite distance forward of the airfoil, there would be no question as
to the initial conditions because the drop would have free-stream velocity
at that distance. However, at a sufficiently large though finite distance ahead of the airfoil, the waterdrop still has essentially freestream velocity. It is then necessary merely to determine this distance
and start the trajectory there. (See section ESTABLISHMENT OF INITIAL
COiXDITIOKS )
-
.
As shown in figure 3 , waterdrops started at different points will
have different trajectories. A waterdrop which has its trajectory
tangent to the upper surface of the airfoil will start at some
NACA RM No. 9AOg
8
position y =
when a large distance ahead of the airfoil. Another
drop at some position y = yoL when a large distance ahead of the airPoil
will have a trajectory which is tangent to the lower aurface of the
airfoil. A l l drops located between YW and 7% at this large distance
ahead of the airfoil will have trajectories which intersect t h e airfoil
surface, that is, the drops w i l l impinge on t h e aurface - specifically on
that portion of the surface limited by the points of tangency of the tangent
trajectories. Au. drops outside you 2 y . 2 ywill miss the airfoil.
As mentioned previously, the area of impingement of waterdrops l i e s
between the point of tangency on the upper surface and the point of
tangency on the lower surface. Mstribution over this area can be
found by determining additional trajectories starting from points intermediate between yolJ and yoL, such as yol and yo2 in figure 3.
The differential analyzer also gives the drop velocities at the
points of impingement. This information is incidental to t h e immediate
purpose of this etudy, but is included with the more pertinent material
in this report for possible future use.
The more important asmptions which it has been necessary to make
in arriving at the simplified probhm which admits of solution are:
(1)A t a large distance ahead of t h e airfoil, the drops move with
free-stream velocity (that is, at the same velocity aa the air) and
with motion parallel to the free-etream path.
(2) The flow of air around the airfoil is that of an ideal fluid
without turbulence or compressibility. (The drag of the air on the
drop is that of a fluid having viscosity.)
(3) The drops are spherical.
(4) No gravitational force acts on the drop.
BSTABUS"
OF INITIAL CONDITIONS
In the study of waterdrop trajectories, the boundary conditions a r e
that the waterdrops are traveling with free-stream velocity at x = -a
(that is, at infinite distance ahead of the airfoil). At finite distances
from the leading edge of the airfoil, the drops have velocity components
and positions varying between those given by the free stream and the
streamlines.
For Airfoil Case I (shown at top in fig. l), the divergence of t h e streamlines is 0.15 percent at x = -3.05, 0.3 percent at x = -2.00, and 1.2 percent
at x = -0.93. Since the divergence is so s m a l l at x zs -3.05 and even
NACA KM No.
9A05
9
f
at x = -2.00,postulating free-stream velocity and position for the d r o p
at x = -2.00 should not 6ause great error in the trajectories. However,
x = -2.00 is too great a distance for obtaining rapid results on t h e
differential analyzer; x = -0.95 being about t h e ma~rimuqpermissible
approach to $he airPail leading edge (for a scale of 20 in. per chord
length on the output table). It waa determined on the analyzer that the
assumption of free-stream values at x = -2.00, for a m a l l and intermediate
0 3
values of .
(2-3, 2 , 2
gave values of y and yd at x = -0.95
which differed from the free-stream values by less than the sxpected
precision of the analyzer, as seen in the following table:
),
The deviation of
from the free-stream value at x = -0.95 is nat
inappreciable but it was determined in the course of the investigation that
the results obtained on the analyzer were the same regardlem of whether id
at x = -0.95 was chosen as the free-stream value or the streamline value.
Further, if choice of free-stream values at x = -2.00 gave values
of y and id at x = -0.95 which were still very close to free-stream
values, then choice of free-stream values at any x further from t h e
airfoil than x = -2.00 would give free-stream values of y and ya
at x = -2.00 since the divergence in streamlines decremes as x
becomes more negative and is a l r e a d y less than the expected precision
of the analyzer at x = -3.05.
For large values of $/F~J(&), choice of streamline values for y
and yd at x = -2.00 resulted in obtaining values of y and id
at x = -0.95 which differed from streamline values by less than the
expected precision of the analyzer as shown in the following table.
NACA FM N O * 9A03
10
6
*/Ru = 2
I
I x = -2.00
(Streamline
values)
Y
0.002
i
-997
9
.00000
I
=
(-‘er)
0.002
9883
0
x = -o:g5 I
I
(Streamline Difference
vdues)
0.002
0
.9882
.0001
0004
- 0004
Hence it would appear that for large values of $&,
t h e initial conditions
should be streamline position and velocity components. However, for large
values of JI/Itv the positions of the drops whose trajectories are tangent
to the upper and lower surfaces of the airfoil, respectively, are quite
close together. At x = -0.93, the distance between the two positions
(measured normal to the free-stream path) choosing streamline conditions
differs by less than the expected precision of the analyzer from the
distance obtained by choosing free-stream conditions.i
On the basis of the above, free-stream values of drop position and
velocity were taken as the initial conditions at x = -0.95 for all
values of $/Rv considered.
For Airfoil Case I1 (shown at center in fig. l), because of the
effect of circulation, it was not possible to assume free-stream conditione
at x = -0.95 for all cams, though the divergence of the streamlines was
about 0.4 percent at x = -2.0 and about 1.4percent at x = -0.8.
Preliminary trajectories were run from x = -0.8 to the airfoil surface
for various values of $/Ru using free-stream co ditione as the initial
conditions. For low values of $/Ru (2-3 and 2 - 9 the choice of freestream conditions as initial conditions seemed appropriate because the
trajectories followed the path of the free stream for about 0.6 chord
length before deviating appreciably and the y-component of velocity of
t h e drop remained equal to the free-stream initial value for about the
same distance. For higher values of $/Ru, the trajectories and y-component
of velocity deviated from the free-stream values almost immediately,
(about 0.1 chord length), indicating that free-stream conditions were not
a suitable choice for initial conditions at x = -0.8.
For these larger values of $&,
the conditians at x = -2.0 were
assumed to be free-stream conditions, and trajectories were run on the
analyzer from x = -2.0 to x = -0.8 for various values of yo and
I
rJACA TIM No. 9A05
li
d,
f o r Jr/%
= 2O,
and 26. From these t r a j e c t o r i e s , the p o s i t i o n and
v e l o c i t y components of a drop a t I = -0.8 were determined as functions
of
and of the p o s i t i o n of the drop a t x = -2.0. These data were
then used aa the starting conditians at x = -0.8 for the determination
of the trac)ectories from I = -0.8 t o t h e points of tangency o r impingement on the a i r f o i l Burface.
+&
Examinatim of the tFa3ectories and y-component of v e l o c i t y p l o t s
= 2O, which were run from x = -2.0 t o x = -0.8 u s i n g freefor
stream i n i t i a l conditions, revealed that t h e r e w a s l i t t l e deviation f o r
about 0.3 chord length, an indication that choice of fme-stream conditions
as i n i t i a l conditions a t x = -2.0 was valid.
Choosing streamline conditions as initial conditions a t x = -2.0
=
for
p v e r e s u l t s which indicated that the waterdrops w e r e
s t i l l following the streauiLine at I = -0.8 (xd = 0.964, f d = 0.0558;
%a= 0 . 9 6 5 , i a = 0.056) and, consequently, that s t r e d i n e c o n d i t i o m
w e r e probably more v a l i d than free-etmam conditlona as i n i t i a l condit i o n s a t x = -2.0. However f o r these l a r g e values of $/Rut the " i n i t i a l "
positions of the drops whoae t r a j e c t o r i e s a r e tangent t o the upper and
lower surfaces of the a i r f o i l , respectively, are quite close together.
A t x = -2.0, the divergence of the streamlines i s about 0 - 3 5 percent,
on t h e average, ( i n the region of the t r a j e c t o r i e s ) so postulating
free-stream velocity and p o s i t i o n a8 t h e i n i t i a l conditions a t x = - 2 . 0
should not introduce too great an error even f o r the l a r g e valuea
Of
'#/Ru.
26
For A i r f o i l Case IV (shown a t the bottom i n fig. l),the "working"
i n i t i a l conditiona, t h a t i s , those t o be used when starting t h e drop
t r a J e c t o r i e s a t x = -0.8, w e r e detexmined by making preliminarg m a
from x = -2.0 t o x = -0.8, (as was done f o r C a m 11) assuming t h e
drops t o have free-stream v e l o c i t y and p o s i t i o n a t x = -2.0.
From these
runs, the p o s i t i o n and v e l c o i t y components of the drop a t r = -0.8
w e r e determined as functions of yo, t h e starting p o s i t i o n a t x = -2.0.
The y-positions of the drops r e l a t i v e t o one another a t x = -2.0 are
t h e same as a t x = -GO under the assumption that the drops have freeThe v a l i d i t y of t h i s
stream v e l o c i t y and p o s i t i o n a t x = -2.0.
JI/Etrr
w
a
s
substantiated,
as f o r Case 11,
assumption f o r most values of
by examination of the t r a e c t o r i e s f o r t h e runs from x = -2.0 t o
x = -0.8. Again, f o r 2-g < Jr/Ru < 26, the t r a j e c t o r i e s followed the
free-stream path f o r about 0.3 chord length before beginning t o deviate
and the y-components of the drop velocity did not change from the f r e e s t r e a m value (0) given them i n i t i a l l y over approximately the same distance.
For higher values of \Ir/%, the choice of streamline conditions as i n i t i a l
conditions seemed more v a l i d because t h e t r a j e c t o r i e s obtained followed
the atreamlines even a t x = -0.8. However, f o r the same reasons given
f o r Case 11, free-stream i n i t i a l conditions vere assumed even a t high
values of $A.
NACA RM NO. 9A05
12
RESULTS
The d i f f e r e n t i a l - a n a l y z e r solutions of t h e equations of motion of the
water”,-=ps were ir: t h e fern ef ~ l o t nnf t h e y-position of the waterdrop as
a function of x and the x-component and y-component of v e l o c i t y of the
waterdrop as a function of x, the distance ahead of the a i r f o i l leading
edge. The y versus x p l o t s were drawn on an output t a b l e with a s c a l e
drawing of the p a r t i c u l a r a i r f o i l mounted at one s i d e of the t a b l e t o
e s t a b l i s h the x and y frame of reference.
(See f i g . 3 . ) In obtaining
t h e tangent t r a j e c t o r y , the analyzer was operated such that a trial
t r a j e c t o r y , s t a r t e d a t some i n i t i a l y-position, yo, was drawn up t o the
v i c i n i t y of t h e a i r f o i l surface. If the t r a j e c t o r y missed the a i r f o i l
surface o r impinged a t some point s h o r t of t h e point of tangency, a new
estimate of the i n i t i a l y-position of the tangent t r a j e c t o r y w a s made
and a second t r a j e c t o r y run on the analyzer. This t r a j e c t o r y w a s usually
close enough t o the tangent one t o permit i n t e r p o l a t i o n (or e x t r a p o l a t i o n ) ,
though occasionally ( i n the f i r s t runs f o r any a i r f o i l ) one o r two more
t r i a l s might be necessary t o determine t h e tangent t r a j e c t o r y s a t i s f a c t o r i l y . Supplementary t r a j e c t o r i e s , with i n i t i a l yo values intermediate
between the values f o r the t r a j e c t o r i e s tangent t o t h e upper and lower
surfaces of t h e a i r f o i l , were run t o t h e i r points of impingement on the
airf‘oil t o permit determination of the d i s t r i b u t i o n of the impingement.
The r a t e of impingement of water on t h a t portion of the surface of
a body bounded by the point of tangency (SL) on the lower surface and
the p o i n t of tangency ( ~ u )
on the upper surface, i a (per u n i t span):
= (You
(=
- Y0L)Uw
-
where Ayo
yoTJ yo,-) is the distance between the i n i t i a l p o s i t i o n s
of t h e upper and lower tangent t r a j e c t o r i e s , U is t h e free-atream velocity., and w i s the l i q u i d water content of the cloud.
Equation ( 9 ) may be rewritten i n terms of an e f f i c i e n c y of water
catch, EM, and t h e maximum catch based on the maximum thickness of t h e
a i r f o i l ( t h a t is, the catch of t h e a i r f o i l when the waterdrop t r a j e c t o r i e s
a r e along the free-stream path), then
MA = UWEM (Maximum thiclmess)
and
(10)
For the intermediate trajectories (see fig. 3 ) , the rate of impingement of water on that portion of the surface of the body bounded by the
point of tangency (sL) on the lower surface and the point of impingement
of the intermediate trajectory (1)is
where
yol
being the initial position of the intermediate trajectory.
From equations (10) and (12)
MA1
MA
-=-
7
E
EM
These equations are the defining equations for the quantities (%, E/%)
which are plotted as functions of RU and 4f and which, with the plots
of q~ and s~ versus Ru, $, permit the computation of the rate and
distribution of impingement of waterdrops on a particular airfoil, given
the necessary data to calculate Ru and 4f.
Tables I, 11, and I11 are summaries of the data obtained using the
differential analyzer for Airfoil Case I, Airfoil Case 11, and Airfoil
Case IV, respectively. The values of xd and yd are the drop
velocities at the points of impingement or tangency. This information
is incidental to the immediate purpose of this study, but is included
with the more pertinent material because of the possible need for it at
some future time. These values of 5, and jTd are reliable except at
high values of $/qT when the velocity components of the drop change
rapidly near the nose of the airfoil.
Figures 4, 9, and 18 are plots of EM, the total percentage catch
versus the Scale Modulus, \c; with the Reynolds Modulus, Ru, as parameter,
for Airfoil Cases I, 11, and IV, respectively. At low values of $, the
curves of constant Ru approach a value of EM which is the m a x i m
is
attainable for the particular airfoil case. This maximum value of
equal to the ratio of the projected frontsl thicbess of the airfoil to
the maximum thickness of the airfoil (17 percent chord in each Airfoil
Case). The values of yo upon which the
values are based are
estimated to be good to 0.0001, as f a r a8 the precision of the differential
analyzer is concerned. Since
is essentially the difference between
two values of yo, a t worst the e r r o r i s about 0.0002. For values of
EM 2 100 percent, the percentage e r r o r i s about 0 . 2 percent b u t f o r
Z 10percent and lower, the percentage e r r o r i s 2 percent
values of
and higher. Hence, a t very high values of q/Ru (Jr/Ru = 26) when there
may be some question of the v a l i d i t y of free-stream conditions as
i n i t i a l conditions a t x = -2.0, the precision of the d i f f e r e n t i a l
analyzer i s such t h a t even i f the correct i n i t i a l conditions had been
used, the percentage e r r o r would s t i l l have been a t least 2 percent o r
higher.
The precision of the t r a j e c t o r i e s could be increased by enlarging
the scale, but then consideration must be made of the runnhg t i m f o r
each t r a j e c t o r y on t h e d i f f e r e n t i a l analyzer. The question of the s c a l e
necessary to give the desired pracision while n o t causing t h e running
time per t r a j e c t o r y t o be excessive i s one which i s posed whether the
i n t e g r a t i o n be performed numerically o r on any kind of computer.
Figure 5 is a p l o t of Su, the distance along the upper a i r f o i l
surface t o t h e point of tangency of the tangent t r a j e c t o r y (that i s ,
the f u r t h e s t point of impingement on the upper surface of the a i r f o i l ) ,
as a function of
with R u as parameter f o r Case I . SL, the
distance along t h e lower a i r f o i l surface t o the point of tangency of
the tsngent t r a j e c t o r y ( t h a t is, the f u r t h e s t point of impingement
on the lower surface of the a i r f o i l ) , i s equal in magnitude t o %
because the a i r f o i l i s symmetrical and a t a = Oo, f o r Case I. A l l
curves of constant Ru approach the value %(= SL) = 0.283, the point
on the surfaca a t which the a i r f o i l has i t s m a x i m thickness, as J,
decreases (waterdrops increase i n diameter). For Case I, there can
be no impingement bsyond t h i s point on e i t h e r surface.
+
Figures 1 0 and 11 a r e p l o t s of S, and SL versus
and R u f o r
Case 11. The maximum value of Su i s now 0-241 and t h a t f o r SL
i s 0.321. These values correspond t o t h e case when +/Rv = 0 ( t h a t i s ,
when there i s no d e f l e c t i o n of the drop by the streamlines).
Figures 19 and 20 a r e similar p l o t s f o r Case IV. The maximum value
of S, i s 0.325 and t h a t f o r SL i s 0.220. A s before, these values
correspond t o the case when there i s no d e f l e c t i o n of the drop by the
streamlines (*/Ru = 0 ) .
The data p l o t t e d i n f i g u r e s 5 , 10, ll, 19, and 20 were obtained by
s c a l i n g off the distances t o the points of tangency on the output p l o t s
of the d i f f e r e n t i a l analyzer. The l o c a t i o n of the exact point of tangency
wa3 not accurately d e t e d n a b l e bscause of the thickness of the i n k - l i n e
representing the t r a j e c t o r y and because of the l a r g e radius of curvature
of both the t r a j e c t o r y and a l s o of the a i r f o i l surface when the t r a j e c t o r y
i s tangent i n t h e region aft of the nose of t h e a i r f o i l . The precision
of location of the points of tangency i s estimated t o be such as t o give
a "maximum e r r o r " of about G . 0 0 2 ( i n terms of chord) a t the lower ends of
t h e curves, W.005 ( i n terms of chord) a t the center, and kO.015 ( i n terma
of chord) a t the upper ends of t h e curves. This maximum error i s n o t a
measure of any inherent error i n the a n a l y z e r t r a j e c t o r y , b u t i s only a
measure of t h e indeterminacy of the l o c a t i o n of the point tangency. The
lowest and highest estimations of the l o c a t i o n of t h i a point were used
i n determining the magnitude of the maximum error and it is probable
t h a t the a c t u a l e r r o r was much less than t h e maximum.
Figures 6 and 7, 1 2 t o 14, and 21 t o 23 are p l o t s of E/% versus s,
the distance along the a i r f o i l surface ( i n terms of chord length) f o r
various values of */% with % as parameter, f o r Case I, Case 11, and
Case I V , respectively. The qjmntity E/%
is t h e r a t i o of the percentage
catch between the point of tangency an the lower surface and any point
of impingement on t h e a i r f ' o i l to the t o t a l percentage catch between the
p o i n t of tangency on t h e lower surface and point of tangency on t h e upper
surface.
.
Figures 8, 15 t o 17, and 24 t o 26 are r e p l o t e of the data of the
preceding paragraph, $/Ru now being the variable parameter and R u being
t h e f i x e d parameter. I n the former f i g u r e s , v a r i a t i o n of Ru with $/Ru
constant did not a f f e c t the d i s t r i b u t i o n greatly (except a t high values
of '#/Ru) but v a r i a t i o n of $/Ru w i t h RU constant changes the d i s t r i bution g r e a t l y f o r all values of Ru, ae shown i n these l a t t e r figures.
The dashed curve f o r $/% = 2-=(lf/Ru = 0) drawn i n figures 15 t o 17 and
24 t o 26 i s based on computed values and i e a l i m i t i n g d i s t r i b u t i o n which
i s obtained when the drops are n o t deflected by the streamlines ( t h a t is,
when the drops a r e very large). This dashed c m e i s n o t drawn i n
f i g u r e 8 because i t almost coincides with the curve dram f o r $/Ru = 2- 6
.
DISCUSSION OF R E m n T S
A s can be seen from reference 5 , the t o t a l percentage catch, EM,
-
t h e mea of Impingement per f o o t of span, Su
SL, and the d i s t r i b u t i o n of
Fmpingement, E/&,
a r e functions of $, the Scale Modulus, and Ru, the
Reynolds Modulus. The range of 9 and RU used in these s t u d i e s
encompasses most combinations of t h e following range of variables:
Variable
Maximum value
Minimum value
2r
U
100
400
20 microns
100 mph
(20,000 f t )
1.267 x 10-3 l b sec2/ft4
3.4 x 10-7 l b s e c / f t *
0.25 f t
1.94 l b sec2/ft4
L
7a
CI
C
yd
(sea x 10'
2.378
3.75 x 10-7
30.0
1.94
16
NACA RM No. 9A05
Figuras 4, 9, and 18, which a r e p l o t s of
versus
with Ru as
parameter f o r Case I, Case 11, and Case m, respectively, show t h a t EM,
based ori the maximum thickness of the a i r f o i l , becomes g r e a t e r than
100 percent when the projected f r o n t a l thickness of the a i r f o i l becomes
g r e a t e r than the maximum thiclmess of t h e a i r f o i l , as it does f o r Case I1
and Case IT. The shape of the curve9 of constant Ru i s t h e same, i n
general, but the slopes tend t o d i f f e r a t t h e upper and lower ends. The
f i v e values of the parameter % f o r Case I bracket t h e f o u r values f o r
Case I1 and Case I V . The reduction i n number of values was d e s i r a b l e
because the number of runs w a s correspondingly reduced while the range
of variables was s t i l l encompassed, f o r the most p a r t . I n order t o b s
able t o compare f i g u r e 4 d i r e c t l y with figures 9 and 18, the dashed l i n e s
f o r the intermediate values of Ru were obtained by i n t e r p o l a t i o n .
Cornsarison of f i g u r e 5 with f i g u r e s 1 0 and ll shows t h a t a t an a n g l e
of a t t a c k of 20, t h e symmetrical 15-percent-thick Joukowski a i r f o i l
e x h i b i t s points of tangency of t h e tangent t r a j e c t o r i e s which a r e c l o s e r
t o the loading edge on the upper surface and f u r t h e r from the leading
edge on the lower surface than f o r the same a i r f o i l a t angle of a t t a c k
of 00, as was t o be expected. Also, comparison of t h e d i s t r i b u t i o n
curves i n figures 6 t o 8 with those shown i n f i g u r e s 1 2 t o 18 shows t h a t
the curves i n the l a t t e r a r e n o t symmetrical about the point s = 0 and,
consequently, t h a t some 60 t o 80 percent of the t o t a l catch impinges on
the lower aurface of the a i r f o i l inertead of the catch being d i s t r i b u t e d
evenly between upper and lower surfaces.
Inspection of the d i s t r i b u t i o n curves f o r Case I V (ftgs. 21 t o 23
o r f i g s . 24 t o 26) shows t h a t from 50 t o 60 percent of the t o t a l catch
impinges on the upper surf'ace of the cambered a i r f o i l except when the
very high. I n t h i s instance, about
drops a r e small and the velocity
60 percent of the catch I s on the lower mrf'ace of the cambered a i r f o i l .
(m)
I n general, the d i 8 t r i b u t i o n curves f o r all three a i r f o i l cases
show t h a t f o r a given value of $/Ru the e f f e c t of varying % is n o t
too g r e a t b u t t h a t f o r a given value of % the e f f e c t of varying $/Ru
i s q u i t e great, t h a t i s , drop s i z e i s r e l a t i v e l y more important than
v e l o c i t y i n determining the d i s t r i b u t i o n of catch.
Figures 27 t o 29 show comparisons of t h e rate of water impingement
f o r the respective a i r f o i l s t o the rate of water impingement on two
cylinders; one with a diameter equal t o twice t h e radius of the leading
edge of the a i r f o i l , the other with a diameter equal t o the maximum
thiclmess of the a i r f o i l (15 percent chord). The comparisons &re made
f o r a low and a high value of Ru. A t high values of $ the former
comgarison i s somewhat b e t t e r , whereas a t low values of $, the l a t t e r
comparison i s much b e t t e r . This was t o be expected because l a r g e
waterdrops ($ low) are not deflected g r e a t l y by the a i r flow and t h e
catch p e r f o o t span is dependent only on t h e projected f r o n t a l thickness
which i s the same i n the l a t t e r instance previously mentioned.
CONCLUSIONS
1. The rates of water impingement on the t h r e e airPoil cases studied
cannot be determined satisfactorily by assuming these rates to be equal
to the rates of water impingement on cylinders except for limited ranges
of $, the Scale Modulus. However, they can be determined within f25 percent
for values of d( between 1 and about 100 or 10,000 (depending upon
by aasuming the rates to be equal to the rates of water
the value of
impingement on a cylinder whose radius is equal to the maximum thickness
of the airfoil.
m),
2. With respect to distribution, the effect of drop size is greater
than the effect of velocity.
3 . Increase of angle of attack of a symmetrical 15-percent-thick,
Joukowski airfoil from a = Oo to a = 2 O , or change from a symmetrical
l?-percent-thick, Joukowski, to EL caoibered, a = 1 mean line, 15-percentthick, Joukowski airfoil, does not change the rate of water impingement
greatly (especially at low Jr and high Ru) but does change the area
of impingement and the distribution of impingement to a greater extent.
~
.
Department of Jbgineering
'-University of California
Lo8 Angeles 24, Calif., October 5, 1948
18
NACA RM No. 9AOg
1. Glauert, Muriel: A Method of Constructing t h e Paths of Raindrops of
Different Diameters Moving in the Neighbourhood of (1)A Circular
Cylinder, (2.) an Aerofoil, Placed in a Uniform Stream of Air; and
a Determination of the Rate of Deposit of the Drops on the Surface
and t h e Percentage of Drops Caught. R. & M. No. 2025,
British A. R. C., 1940.
2. Kantrowitz, Arthur: Aerodysnamic Heating and the Deflection of Drops
by an Obstacle in a n Air Stream in Relation to Aircraft Icing.
NACA TN No. 779, 1940.
3. Bergrun, Norman R.: A Method for Numerically Calculating the Area
and Distribution of Water Impingement on the Leading Edge of an
Airfoil in a Cloud. NACA 'I" No. 1397, 1947.
4. Langmuir, Irving, and Blodgett, Katherine B.: A Mathematical Investigation of Water Droplet Trajectories. General Electric Co. Rep., 1943.
(Also available as Army Air Forces Tech. Rep. No. 5418 and as
Dept. of Commerce Pub. €9 No. 27565.)
5. Boelter, L. M. K., Young, George, and Tribus, Myron:
The Limitations
and Mathematical Basis for Predicting Aircraft Icing Characteristics
from Scale Model Studies. Section V of Army Air Forces Tech. Rep.
NO. 5529, NOT. 6, 1946, pp. 63-78.
TABLE I
4/27/48
1-1-3-1
4/27/48
1-1-3-1
0
15sThlok
-trloal
a- 0
1
1
I
I
i
i
i
i
i
I
1
I
I
i
i
i
i
i
2l
27
0.074
upper” . 0.265
1.0
0
0.148
1.0
-0.074
Lover+
-0.265
1.0
0
0
0
0.273
0 . 9 ~ 0.004
4/22/48
1-1-4-2
Z3
Z9
0.074
Upper’
4 f22 /48
1-1-4-2
23
29
-0.074
Lower+ - 0 . 8 3
0.99
0.148
1.0
-0.W4
0
0
1.o
I
4/22/48
4/22/48
1-1-5-3
25
211
0.072
Upper’
0.262
0.m
0.013
O.IW+
1-1-5-3
25
211
-0.072
Lover+
-0.62
0.997
-0.013
0
1-2-2-1
22
25
0.073
Uppei*
0.273
1.0
0.012
0.146
1-2-2-1
22
e5
-0.073
Love+
-0.273
1.0
-0.012
0
0 .
1.o
4
0
1-2-3-1
24
27
0.070
Upper’
0.244
1.005
0.023
0.140
1.o
1-2-3-12
24
27
0.045
Upper
0.068
0.9
0.013
O.ll5
0.821
1-2-3-11
24
27
0.020
Upper
0.021
0.962
0.009
0.Op
0.643
1-2-3-11
24
27
-0.020
Lover
-0.021
0.982
-0.009
0.050
0.357
1
0.933
I
I
4/23/48
1-2-5-21
I 2’ I 9I 0.020 I Upper I 0.023 I 0.931 1 0.029 I 0.078 I
.
XTangent T r a j e c t o r y .
0.672
I
~~
~
NACA RM No. 9AO5
20
TABLE I
- Concluded
YATWWOP TRAJECTORY VALUE O B T m Pam D
Smtrlcal
m
I
A
L AApLyzw FOR JOUKOWSKI AIREUIL,
15%Thick
a
Date
Run No.
9
Ro
yo
4/30/48
RR
1-4-1-3
26
Z3
0.0255 u p p +
0.321 0.0510
0.078 0.870 -
4/21/48
1-4-1-22
$
23
0.018
Upper
0.031 0.693
4/21/48
1-4-1-21
$
Z3
0.008
Upper
0.010 0.69e
a .061
Lower
-0 . O ~ O 0.698
-0.061
Lower
-0.19
-0.031 0.693 -
Love+
-0.078 0.870 -0.321
5/4/48
FR
1-4-4-1
212
Z9
I
I
0'
a .iw 0 .Ob35
0 J3335
o .657
- 0.0175
0.343
0.147
0
0
I
Upperi
0.073
0.628
0.042
Love+
-0.073
0.628
0
Uppe+
0.050
0.681
a A76
0.0290
1.o
Uppar
0.020
0.572
0.198
0.0245
0.a5
1-0
0
Upper
I 0.009 10.563 I 0.109 I0.0195 I
0.672
Lover
-0 .Om 0.572
o. ~ 4 5
0 J55
Lover*
-0.050
0.681 -0.476
Upperi
0.052
0.741
Lovelx
-0.052
0.741 -0.451
~
.
I
~
I
-0.198
0.280
0 J93
I
0
0
1.o
0.451 0.030
0
0.200
0
~
I
I
I
1 0.022 10.329 I 0.469 IO.0080 I
Love+
I -0 .022 I O .329 I -0.469 I 0
I
I
I
I
I
Upper,
I
I
1.o
0.053
0
I
I
NACA RM NO. 9 ~ 0 5
21
TABLE I1
NACA RM No. 9AO5
22
TABLE I1 - Concluded
Spmetrical
I
a = 2'
15% Thick
6/30/48
2-3-4-2U
21°
21°
-0.0743
Upperf
0.063
0.914
0.424
0.0617
1.Oo0
6/30/48
2-3-4-2L
21°
21°
-0.1360
Lower* -0.118
0.902
-0.195
O.oo00
O.Oo0
I 2-4-2-2U I I 26 I-0.1005 I Upper* I 0.038 10.767 I 0.528 1 0.0377 I
7/1/48 I 2-4-2-lD 129 1 Z6 )-0.1080I Upper I 0.009 10.553 1 0.303 I 0.0297 I
7/1/48
g9
1.ooO
0.603
7/7/48
2-5-2-9
212
26
-0.1243
Upper*
0.003
O.Og0
0.434
0.0035
1.ooO
7/7/48
2-5-2-5
212
26
-0.1278
Lower*
-0.015
0.246
-0.235
b.OKXl
O.Oo0
7/7/48
2-5-3-4U 214
28
-0.1254
Upper*
.O.oOe 10.103
0.506
0.0021
1.000
7/7/48
2-5-3-4L
28
-0.1275
Lower* -0.014 10.175 -0.295
O.oo00
O.Oo0
214
---I
*T'Tangent Trajectory.
0.251
0.802
I 2-4-2-ZD 129 I 26 1-0.1155 I Lower I -0.004 10.542 I 0.132 1 17.0232 I
7/1/48
0.411
0.023
0.014
NACA RM No. 9A05
23
TABU m
b
.
7/13/48 4-2-3-2U 25 '2
0.0775
Upper'
0.275
L O P
0.0%
0.1375
lxxxl
7/13/48 4-2-3-u)
0.0503
Uppr
0.092
O.g@
0.042
0.m3
0.8M
0.0225
Upper
0.034
0.g6
0.09
O.Ote5
0.600
25 28 - 3 . ~ 4 5 - Upper
0.001
0.92
0.0017
0.0555
0.404
25 '2'
7/13/48 4 - 2 - 3 4 25
7/13/48 4-2-3-9
Z8
0.1005
0.8J7
0.0745
0.598
0 . 0 ~
0.402
0.02%
0901
o.oo00
0.000
0 -917
I
.
7/15/40
4-3-3-20
1 Upper
Z8 2'
-0.0165
7/15/48 4-3-3-9
'2
2'
-0.03b
Lasr
7/15/48
2'
'2
-0.0510
Lasr
4-3-3-4D
7/15/48 4-3-34,
28 28 -0.068~
b r +
0.1147
1.000
o.oo00
oiwo
0.765
NACA RM No. 9 A 0 5
24
T
W I11 - Concluded
WA'ENBOP TRAJECTORY VALUES CBTAINEU FROM DIFFWmIAL ANALYZER FOR JOUKOWSKI AIKFOIL,
Cambered
7 16 48 44-2-31
29
Tangent Trajectory.
a = 1 Mean Line
15% Thick
a = Oo
NACA RM No. 9A05
25
.
0
C
II
II
3
3,
)
3
II
d
3
h
x0
4
4
0
g
8
h
xc)
Yl 1
0
$
I
a,
a,
h
k
a,
a,
k
jI!i
l
a,
a,
Fr
k
26
Drop velocity
‘d
Figure 2.-
+X
Diagram of velocity components of air stream and waterdrop.
7d
3
G
Q)
k
Q)
2
a
Q)
9
w
0
rn
4
0
4
a
4
3
a
4
3
0
E
0
k
w
G
9
4
cd
cd
4
a
d
cd
0
EE-l
I
c6
.
2
.I+
k
NACA RM No.
28
0
4
0
9A05
us
.r(
Frc
.
8
0
us
c?
0
?
0
us
0
In
1
1
0.
0
0
0
0
30
NACA RM No.
9A05
32
NACA RM No. 9A05
33
,
NACA RM N o .
34
In
c‘?
0
*.
In
1
0
1
0
9A05
.