APPENDIX A
NACA TN 2952 APPENDIX A METHOD USED TO CALCULATE INCOMPRESSIBW F L O W FIELD AROUND AI;RFOIL The v e l o c i t i s i q t h e two-dimensional flow f i e l d were calculated b: d i s t r i b u t i n g a sheet of vortices on t h e a i r f o i l surface of such strength t h a t the v e l o c i t i e s on t h e surface caused by the vortices were t h e same N as t h e v e l o c i t i e s measured i n a wind tunnel. The principles upon which a pc t h i s method i s based are established i n reference 11.
N The velocity a t the surface of an a i r f o i l can be determined from a knowledge of the pressure coefficient Cp and the free-stream Mach nmiber M with the a i d of the following expression: The pressure coefficients f o r a large n W e r of points on the surface of t h e NACA 6S1-208 and 651-212 a i r f o i l s a t an angle of a t t a c k of 4 ' were obtained from wind-tunnel data taken a t the NACA Ames laboratory f o r several free-stream Mach numbers. The surface velocities used t o c a l - culate the flow f i e l d are shown i n figure 11 f o r a Mach nurober of 0 . 2 .
The flow f i e l d s a t other Mach numbers were not calculated, because the r e s u l t s presented i n reference 5 show t h a t the e f f e c t of the compressi- b i l i t y of t h e air on the droplet t r a j e c t o r i e s is negligible.
The velocity a t a point i n a flow f i e l d caused by an element of the vortex sheet of strength fl placed a distance r (?fig. 12) from the' point is If an element of vortex sheet of strength i s placed on an increment of the a i r f o i l a t the ith position on the A3 a i r f o i l ' s u r f a c e , the velocity caused only by the ith section of t h e a i r f o i l is
(.e> i
uv = 2 n r i 16 BACA TN 2952 a t a point i n the flow f i e l d a t a distance ri from the ith section ( f i g . 12). The l o c a l components of the perturbation velocity, a t a point i n the flow f i e l d , caused by 300 vortex elements distributed on both the upper and lower surfaces of the a i r f o i l are s Y'-9 &i -LtO The coordinate system shown i n figure 1 2 differs from that shown i n f i g - ure 1, i n that i n figure 1 2 one of the coordinates coincides with the geometric chord l i n e . The direction of the free-stream velocity is toward the a i r f o i l a t an angle a w i t h respect t o - X I . The horizontal
and v e r t i c a l components of the l o c a l velocity % I and uyl, respec-
tively, a r e obtained by adding V cos a t o u x l J V and V s i n a t o
The coordinate system shown i n figure 12 was used because of its u Y f r V - adaptability t o available calculating equipment and procedures. The t r a j e c t o r i e s were solved i n t h e primed coordinate system shown i n f i g - ure 1 2 and l a t e r transformed graphically t o the system presented i n f i g - ure 1.
A t o t a l of 300 vortex elements were used on the a i r f o i l with a much denser d$strib.ution on t h e forward section than beyond the 50-percent-
Equations (a) were solved with electronic calculating
chord point.
machines. Approximately 300 points were computed i n 6he flow field out t o 1 chord 1ength.ahead of the a i r f o i l i n the region of i n t e r e s t with regard t o computing the t r a j e c t o r i e s of droplets t h a t s t r i k e the a i r f o i l .
Between 1 and 5 chord lengths ahead of the a i r f o i l leading edge, the flow-field velocity components were approximated by assuming t h a t the flow was caused by a single vortex located on the a i r f o i l chord line The strengkh of this 25-percent chord inward from the leading edge.
vortex was determined by the requirement t h a t a t x' 9t -1 the v e r t i c a l velocity caused by t h i s single vortex m u s t be the same as t h a t complted with equation (A2)
APPENDIX B
NACA TN 2952 APPENDIX B GRAPRICAL PROGEDURE F O R TRANSLATION O F PRACTICAL FLIGHT
CONDITIONS IN TERMS O F DIMENSIONUSS P - T E R S
A graphical procedure i s presented t o a i d i n t h e t r a n s l a t i o n of N airplane speed, chord s i z e , a l t i t u d e , and droplet diameter i n t o terms of t h e dimensionless parameters K and R e o used i n t h i s report. A N solution of equation ( 2 ) i s presented i n f i g u r e 1 3 f o r two altitudes.
For given droplet diameters in.microns and r a t i o s of t h e chord length i n feet t o t h e f l i g h t speed i n miles per hour, the reciprocal of t h e i n e r t i a parameter can be determined a t a l t i t u d e s of e i t h e r 10,000 o r 30,000 f e e t from figure 13. Altitude does not appreciably a f f e c t t h e of 1/K, as can be noted from a comparison of values i n f i g - value ure 13(a) with those i n f i g u r e 13(b).
A n a i r f o i l with a 12-foot chord length a t a f l i g h t speed of 400 miles per hour and an a l t i t u d e of 10,000 f e e t passing through a cloud composed of droplets a l l of which are 1 7 microns i n diameter w i l l ’ be used a s an example i n t h e graphical procedure t o i n t e r p r e t p r a c t i c a l f l i g h t units i n t o terms of the dimensionless parameters. The value of
- = - - - 0.0300
1/K is obtained from figure 13(a) f o r t h e value of U 400 and droplet diameter d = 1 7 . The value of l / K obtained from f i g - ure 13(a) is 21.
The free-stream Reynolds number f o r d i f f e r e n t a l t i t u d e s may be obtained from figure 14. The product of t h e droplet diameter i n microns The Reynolds and the flight speed i n miles per hour must be known.
number i s a function of t h e air density, which depends on the pressure and the temperature a t t h e a l t i t u d e s considered. The pressure used t o calculate t h e air density w a s taken from t a b l e s of NACA standard atmos- pheric pressure a t various altitudes, but t h e temperature w a s based on t h e most probable icing temperature a t various a l t i t u d e s . The most probable i c i n g temperature w a s obtained from approximately 300 i c i n g observations (ref. 1 5 ) and i s presented i n figure 15. For the example under consideration, the product of the droplet diameter and the f l i g h t Reo, obtained from f i g u r e 14, The value of speed i s (17) (400) = 6800.
i s 171- The following r e l a t i o n s are presented f o r use when the degree of The accuracy required is not attainable with the graphical procedure.
values f o r t h e viscosity p should be obtained from f i g u r e 16; these of figure 15.
values are based on the most probable i c i n g temperature The charts of figures 13 and 14 are based on the most probable i c i n g temperature and viscosity.
18 IJACA TIq 2952
-12 d u
K = 1-704X 10 -
PL
d =D 7.662X105 @
P
pa =D 0.0412 -
Ta where the units are given in t h e l i s t of synibols. (The density of water w a s assumed t o be 62.46 lb/cu f t and t h e acceleration due t o gravity,
32.17 ft/sec2 - )
RETERENCES 1. Glauert, Muriel: A Method of Constructing t h e Paths of Raindrops of Different Diameters Moving i n the Neighborhood of (1) a Circular Cylinder, (2) an Aerofoil, Placed i n a Unifomn Stream of A i r ; and a Determination of t h e R a t e of Deposit of the Drops on the Surface and t h e Percentage of Drops Caught. R. & M. No. 2025, B r i t i s h A.R.C., 1940.
2 . Ranz, W . E. : The Impaction of Aerosol P a r t i c l e s on Cylindrical and Spherical Collectors. Tech. Rep. No. 3, Eng. Exp. Station, Univ.
Ill., March 31, 1951. (Contract No. AT(30-3)-28, U.S. Atomic Energy Commission.)
3. Langmuir, Irving, and Blodgett, Katherine B.: A Mathematical Inves- t i g a t i o n of Water Droplet Trajectories. Tech. Rep. No. 5418, A i r
Materiel Command, AAF, Feb. 19, 1946. (Contract No. W-33-038-a~-
9151 with General E l e c t r i c Co.)
4. Brun, Rinaldo J., and Mergler, Harry W . : Impingement of Water Drop- l e t s on a Cylinder i n an Incompressible Flow F i e l d and Evaluation of Rotating Multicylinder Method f o r Measurement of Droplet-Size Distribution, Volume-Median Droplet Size, and Liquid-Water Content i n Clouds. NACA TN 2904, 1953.
5. Brun, Rinaldo J., Serafini, John So, and Gallagher, Helen M.: Impingement of Cloud Droplets on Aerodynamic Bodies as Affected by the Compressibility of A i r Flow Around t h e Body.
NACA TN 2903, 1953 * NACA TN 2952 19
6. B r u n , Edmond, Caron, Robert, e t Marcel, Vasseur: Introduction k
llktude de la M&canique des Suspensions. G.R.A. Rapport Tech..
No. 15, Recherches Agronautiques (Paris) , 1945 a
7. Bergrun, Norman R.: A Method f o r Numerically Calculating the Area and Distribution of Water Impingement on the Leading Edge of an A i r f o i l i n a Cloud. NACA TN 1397, 1947.
8. Guibert, A. Go, Janssen, E., and Robbins, W. M.: Determination of
'0 4 R a t e , Area, and Distribution of Impingement of Waterdrops on r, u Various Airfoils f r o m Trajectories Obtained on the D i f f e r e n t i a l Analyzer. NACA RM 9A05, 1949.
9. Bergrun, Norman R.: An Emperical Method Permitting Rapid Determina- t i o n of t h e Area, R a t e , and Distribution of Water-Drop Impingement on an A i r f o i l of Arbitrary Section a t Subsonic Speeds.
NACA TN 2476, 1951.
10. Brun, Rinaldo J., Serafini, John S., and Moshos, George J.: kl Impingement of Water Droplets on an NACA 651-212'Airfoil a t an Angle of Attack of 4'. NACA RM E52B12, 1952.
Q 2
Y) I 11. von Mises, Richard: Theory of Flight. McGraw-Hill Book Co., Inc., 3 F i r s t ea., 1945.
12. Gray, V. H., Bowden, D. T., and von Glahn, U. : Preliminary R e s u l t s of Cyclical De-icing of a Gas-Heated Airfoil.
NACA RM E51J29, 13. Lewis, James P., and Bowden, Dean T. : Preliminary Investigation of Cyclic De-Icing of an A i r f o i l Using an External E l e c t r i c Heater.
NACA RM E51J30, 1952.
14. Dorsch, Robert G., and Brun, Rinaldo J.: A Method f o r Determining on Swept Wings. NACA mT 2931, 1953.
Cloud-Droplet Impingement 15. Hacker, Paul T., and Dorsch, Robert G.: A Summary of Meteorological Conditions Associated with Aircraft Icing and a Proposed Method of Selecting D e s i g n Criterions f o r Ice-Prevention Equipment.
NACA TN 2569, 1951.
NACA "PI 2952 P I II x NACA T N 2952 NRCR TN 2952 Ciiord 1 ength; 0 100 200 300 400 500 600 Flight speed, mph (a) Droplet size, 15 microns.
3. - Rate of water impingement. Angle of attack, 4 ' ; alti- Figure tude, 20,000 feet; most probable icing temperature, -1l0 F.
NACA TN 2952 23 Chord length, c, $
%
M r: rl ! a E rl k a , c, Kl k a i c, l d k rl Kl c, m 0 200 300 40 0 500 600 Flight speed, mph (b) Droplet size, 20 microns.
Figure 3. - Continued. Rate of water impingement. Angle cf attack, 4 ' ; altitude, 20,000 feet; most probable icing tnmper- ature, -no F.
NACA "N 2952 Plight speed, mph (0) Droplet size, 30 microns. (d) Droplet size, 40 microns.
.
Angle Of attack, 4'; altitude, 20,000 feet; most probable icing F i g u r e 3 . - Concluded. Rate of water impingement.
temperature, -11O P.
NACA TN 2952 N ZD c - N 0 5 10 15 20 25 c Pressure a l t i t u d e , f t Figure 4. - E f f e c t of a l t i t u d e on r a t e of water impingement.
Chord length, 10 f e e t ; a i r p l a n e speed, 300 miles per hour; most probable i c i n g temperature, (see appendix B ) .
HACA T1\s 2952 0 d * m c NACA T N 2952
L 9 9
d N r l m . ?
NACA TN 2952 N -3 0, N .06
3 .05
r/f
-
a) cd k k .04 F l i g h t speed, mph (a) Droplet s i z e , 15 microns.
F i g u r e 6. - L i m i t of impingement along upper s u r f a c e . A l t i t u d e , 20,000 f e e t ; angle of a t t a c k , 4O; most probable i c i n g temper- a t u r e , -11O F.
NACA TN 2952 29 Plight speed, mph ( b ) Droplet s i z e , 20 microns.
Figure 6. - Continued. L i m i t of impingement alorig upper surface.
Altitude, 20,000 f e e t ; angle of attack, 4O; most probable i c i n g temperature, -11O F.
30 NACA TN 2952 - 651-208 0 100 200 300 400 500 600 speed, mph (d) Droplet size, 40 microns.
Altitude, 20,000 feet; angle of attack, 4'; most probable icing NACA TN 2952 N D c N 32 NACA TIV 2952 C r F l i g h t speed, mph ( c ) Droplet size, 30 microns.
Figure 7. - Continued. L i m i t of impingement along lower surface.
Altitude, 20,000 f e e t ; angle of a t t a c k , 4O; most probable i c i n g temperature, -1l0 F.
NACA TN 2952 33 .5t .5; * 4i .41 f.
bD E rl a 8 .4( 0 c, 0 d 0 . 3 &i ,.
rn
$ .3
i
rl d
:: .2
d b
z
rl .2 &2 E E, M c .2 d d c u d .1 =: .1 .o .o I Flight speed, mph (d) Droplet size, 40 microns.
Figure 7. - Concluded. Limit of impingement along lower surface.
Altitude, 20,000 feet; angle of attack, 4O; most probable icing temperature, -1l0 F.
NACA TN 2952 i 3 4-1 I a3 NACA TN 2952 N a r- cu 36 NACA T ! N 2952 NACA TN 2952 r-i m tb -4 0
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I HACA TN 2952
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a J k a , l-i r M) r: C (d c .h ri rl c, k d Ld NACA TN 2952 39 d (D k rl 0 a . h 0 s 0 - 9 hD
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d
k (It l-i aJ I 4 a
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d I 40 NACA TN 2952 NACA TN 2952 NACA "H 2952 rl NACA TN 2952 NACA TN 2952 N - 3 I c n N
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NACA TI? 2952 o w w d N d 0 0 0 0 0 0 0 0 d c U , . i 3 d d 2 W W d (\I 0 w w 6 - 4 . 4 r l 3 NACA TN 2952
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NACA TN 2952 47 Altitude, f t Figure 14. - Free-stream Reynolds number as function of a l t i t u d e , NACA TN 2952 NACA TN 2952 49 K) rl X M M M M