Impingement of Water Droplets on an NACA 65(sub 1) -212 Airfoil at an Angle of Attack of 4 Deg

RM E52B12

NACA

RESEARCH MEMORANDUM

IMPINGEMENT OF WATER DROPLETS ON AN NACA 65 - 212 AIRFOIL AT AN ANGLE OF ATTACK OF By Rinaldo I. Brun, John S. Serafini and George I. Moshos Lewis Flight Propulsion Laboratory Cleveland, Ohio

NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

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WASHINGTON September 10, 1952 file$ Of the for Aeonaw..

/ / *shhigton, U, C.

7'Z ERRATA NO.

/e /e s e 0w, NACA EM E52B12 IMPINGFMFNT OF WATER DROPLETS ON AN NACA 6 51-212 AIRFOIL AT AN ANGLE OF ATTACK OF 1400 By R. J. Brun, J. S. Serafini, and G. J. Moshos September 16, 1952 The ordinate values in figure of this report should be divided by 10.

NACA RM E52B12 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEI.VRANDUM IMPINGEMENT OF WATER DROPLETS ON AN NACA 651-212 AIRFOIL AT AN ANGLE OF ATTACK OF 40 By Rinaldo J. Brun, John S. Serafini, and George J. Mshos SUMMARY The trajectories of droplets in the air flowing past an NACA 651-212 airfoil at an angle of attack of 40 were determined. The collection efficiency, the area of droplet impingement, and the rate of droplet impingement were calculated from the trajectories and are presented herein to cover the following range of conditions: Variable Minimum value £v.ximum value Droplet diameter (microns) 5 100 Airplane speed. (mph) 150 Critical flight speed 1000 35,000 Altitude (ft) Chord length (ft) INTRODUCTION As part of a comprehensive research program directed toward an appraisal of the problem of ice prevention on high-speed aircraft, an investigation of the impingement of cloud droplets oil airfoils and other aerodynamic bodies has been undertaken at the NACA Lewis laboratory.

The investigation includes a study of the extent of impingement on a low- drag airfoil and the rate of droplet impingement per unit area of the airfoil area affected. Previous investigators have calculated the water- droplet trajectories for cylinders (references 1 and 2) and for Joukowski airfoils (references 3 and 4). An empirical method for determining area, rate, and distribution of water-droplet impingement on airfoils of arbi- trary sections is presented in reference 5. The method is more firmly established for 15-percent-thick airfoils resembling Joukowski airfoil sections than for low-drag airfoils, because the basic data used in developing the empirical method were obtained for four Joukowski airfoil sections and only one low-drag section. Further. water-droplet trajectory data are needed for low-drag airfoils and particularly for airfoil sections thinner than the 15-percent-thick sections for which results are reported in the references cited.

NACA RN E52B12 The studies presented in thiiB report are for an NACA 651-212 airfoil, which is a 12-percent-thick wing, placed at an angle of attack of 40• The NACA 65-series airfoil sections are particularly adaptable to air- planes having high-level flight speeds. An airfoil 12 percent thick was chosen as adaptable to transport and cargo airplanes. An angle of attack of 40 was chosen as being representative of low cruise attitude for a turbojet-powered aircraft operated under conditions giving a relatively large area of droplet impingement on the airfoil. The results presented herein are applicable to the NACA 651-212 airfoil under the following conditions: chord lengths from 2 to 20 feet; altitudes from 1000 to 35,000 feet; airplane speeds from 150 miles per hour to the critical flight Mach number; and droplet diameters from 5 to 100 microns.

ANALYSIS As an airfoil moves through a cloud, the interception of the cloud droplets by the airfoil is dependent on the physical configuration of the airfoil and on the inertia of the cloud droplets. In order to obtain the extent of impingement and the rate per unit area of droplet impinge- ment on an airfoil, the cloud droplet trajectories with respect to the airfoil must be determined. The differential equations that describe the droplet motion have been stated in reference 2 and are presented herein in the following form: dvx CDRe1 u - v) = 24 R ( (1) dvYCDRe1 (uy_vy) dT - 24 where 2 U w a (2) K 9 p.L and (Re (3) = ( U X - vx) 2 + (uy - vy) 2 (All symbols are defined in appendix A.) The dimensions of the free- stream velocity U in equation (2) are feet per second in order that the dimensions of the other variables in the equation be consistent with the definitions given in the list of symbols.

NACA RM E52B12 The differential equations (1) state that the motion of a droplet is governed by the drag forces imposed on the droplet by the relative motion between the droplet and the air moving along the streamlines around the airfoil. The droplet momentum tends to keep the droplet moving in a straight path, while the drag forces tend to force the droplet to follow the streamlines. For very small droplets and slow speeds, the momentum of the droplets parallel to the direction of the free-stream motion is small and the drag forces are large enough that little deviation from the streamlines occurs; whereas for large droplets and high speeds, the momentum is great enough to cause the droplets to deviate from the streamlines. In accordance with equations (1) and the definition of the parameter K in equation (2), for a given size and configuration of air- foil, the trajectories depend on the radius of the droplets, the air- speed, and the air viscosity as first-order variables. The trajectories also depend on the physical configuration of the airfoil and its angle of attack, in that these two variables determine the magnitude of the and Uy everywhere in the flow component velocities of the air UX field.

The component air velocities were determined by a vortex substitution method that requires a knowledge of the pressure distribution on the surface of the airfoil. The pressure distribution was obtained from wind- tunnel data taken at the Ames laboratory. The method for calculating the local perturbation velocities in the two-dimensional incompressible flow field ahead of the airfoil is described in part in reference 6 and is presented more fully in appendix B. The computations were performed with electronic calculating machines employing punched cards. An incompress- ible flow field was obtained with the vortex substitution method. Pre- liminary calculations showed that the effect of compressibility of air on the trajectories of droplets was not a first-order effect up to the critical Mach number of the airfoil. Because of these preliminary cal- culations, the results presented herein are applicable up to the flight critical Mach number.

The more important assumptions that have been necessary in order to solve the problem are: (1) At a large distance ahead of the airfoil (free-stream.conditions) the droplets move with the same velocity as the air.

(2) The droplets are always spherical and do not change in size.

(3) No gravitational force acts on the droplets.

NACA RN E52B12 METHOD OF SOLUTION The differential equations of motion (equation (i)) are difficult to solve by ordinary means because the values of the velocity components of the air and the term containing the coefficient of drag are not known until the trajectory is traced. These values are determined as the trajectory of a droplet is developed because the magnitudes depend on the position of the droplet in the flow field. Simultaneous solutions for the two equations were obtained with tie use of a mechanical analog constructed at the Lewis laboratory for this purpose. The answers were obtained in the form of plots of the droplet trajectories with respect for the droplets, required to the airfoil. The coefficient of drag CD in equations (1), was obtained from tables in references 2 and 3.

The equations of motion (equations (i)) were solved for the following values of the parameter K: i/ioo, 1/50, 1/10, 1/5, 1, and 2. For six each value of the parameter K a series of trajectories was computed for each of three values of free-stream Reynolds number Re 0 (16, 256, and 1024). (A graphical procedure for interpreting the dimensionless parameters used in this report in terms of airplane speed, chord length, altitude, anddroplet size ispresented in appendix C.) Each series of trajectories encompassed the airfoil with a trajectory that was tangent to the upper surface of the . airfoil and with a trajectory that was tangent to the lower surface of the airfoil. The upper and lower tangent and trajectories started at free- . stream conditions at distances YO,u YO , respectively, below the geometric chord line of the airfoil (fig. 1). The geometric chord line of the airfoil is oriented to coincide with the x-axis of the rectangular coordinate system; and the leading edge is placed at the origin of the coordinates. At an infinite distance ahead of the airfoil, the unifbrm air flow carrying the cloud droplets is assumed to be approaching the airfoil from the negative x-direction at an angle of 40 with respect to the geometric chord line. All distances are dimensionless because they are ratios to the airfoil chord. length L, which is assumed to be the unit of distance.

Before the integration-of the equations of motion could be performed with the analog, the initial velocity and acceleration of the droplets had to be determined. As postulated in the assumptions, at an infinite distance ahead of the airfoil, all the droplets have vertical and hori- zontal components of velocities that are the same as those of the free- stream air. At finite distances ahead of the leading edge of the airfoil, the droplets have velocity components and positions varying between those pertaining to the free stream and to the streamlines. At 5 chord lengths ahead of the airfoil, the air streamlines were found to deviate from free-stream conditions by less than the expected accuracy of the analog; therefore this point was assumed to be a safe position to assign the initial conditions to the droplets as being those of the

NACA RM E52B12 air streamlines. Because the time required to trace each trajectory from 5 chord lengths ahead of the airfoil up to the airfoil surface was prohibitive, the plotting by the analog was started at 1 chord length ahead of the airfoil leading edge. The starting conditions at 1 chord six length ahead of the leading edge were determined for each of the values of K studied by calculating a sample trajectory that started at x = -5. A preliminary study showed that while a droplet is approaching a position 1 chord length ahead of the airfoil, the amount of deviation of the droplet trajectory from the air streamline on which the droplet had started depends only on the value of K and not on the starting value of y at x = -5, provided the value of y is within the region of interest in this problem. Because the sample trajectories for each value of K studied were calculated from x = -5 to x = -1 in order to determine the starting values of droplet velocity and y- ordinate at x = -1, the final results were the same as if each trajectory were calculated from 5 chord lengths ahead of the airfoil leading edge.

A study also revealed that the assignment of either free-stream or streamline values to the droplets at x = -1 was not sufficiently accurate, because the trajectories from 1 chord length ahead of the air- foil surface are very sensitive to small variations in the droplet velocity starting conditions assigned at 1 chord length ahead of the air- foil.

RESULTS AND DISCUSSION The series of trajectories computed for each combination of values of K and Re 0 studied permits the evaluation of the area, the rate, and the distribution of water-droplet impingement on the NACA 651_212 airfoil section at an angle of attack of 4 0 . The tangent trajectories determine the area, or extent, of impingement. All droplets having trajectories between the tangent trajectories will strike the airfoil, whereas all droplets having trajectories not bounded by the tangent trajectories will miss the airfoil. The tangent trajectories also deter- mine the rate of over-all droplet impingement, because the amount of water-droplet impingement on the whole wing is governed by the spacing fig. i) at a large distance of the tangent trajectories ( you - y01) ahead of the airfoil where the cloud is uniform. The manner in which all the droplets collected are distributed over the area of impingement is determined by the behavior of those trajectories that are boundedby the tangent trajectories. - The results are often presented herein as functions of the param- eter K. The parameter K has been called the inertia parameter, because its magnitude directly reflects the external force required on a droplet to cause a deviation from its original line of motion. For large values of K (that is, K> 1) which correspond, for example, to

6 NACA RM E52B12 droplets larger than 50 microns in diameter moving toward a 6-foot-chord airfoil at 400 miles per hour, the droplet trajectories deviate by only small amounts from straight lines. For values of K less than 1/50 which correspond, for example, to droplets less than 12 microns in diameter moving at less than 300 miles per hour toward an airfoil section with a 12-foot chord, the droplet trajectories more nearly coincide with the air streamlines.

The results are also often presented in terms of the scale parameter Re 9L 0 = a (4) - K ap 4r where L and a must be in the same units. The parameter is called a scale parameter, because at any given altitude the value of i varies directly as the ratio of the airfoil chord length to the droplet size.

For a tapered wing, the parameter permits the direct evaluation of '1' the amount of spanwise droplet impingement on the wing, because the entire wing is subjected to droplets of the same size and to the same air and water densities. The chord length is the only spanwise variable appearing in the scale parameter. For each section of span considered, the taper must be small enough that two-dimensional flow over the section is approximated. The scale parameter also permits a direct comparison of the impingement that can be expected on an airfoil passing through numerous clouds each composed of droplets uniform in size in each cloud but varying in size from one cloud to another.

Rate of water interception. - In flight the total rate of water interception, in pounds of water per hour per foot of wing span, is deter- mined both by the droplet trajectories and by the meteorological condi- tions. The liquid-water content, in grams per cubic meter, and the, droplet size are the important meteorological conditions. The speed and the size of the airfoil, as well as the droplet size, affect the tangent droplet trajectories, which determine the droplets that strike the airfoil. The total rate of water interception per unit span of the air- foil on that portion of the airfoil surface bounded by the upper and the lower tangent trajectories (fig. 1) can be calculated from the in±'orma- tion in figure 2 and the following relation: ) LUw cos a - y0 (5) Wm = 0.329 where the flight speed U is in miles per hour. Figure 2 is a plot of the spacing between the upper and the lower tangent trajectories at free- stream conditions as a function of the reciprocalof the inertia param- eter.

NACA EM E52B12 The rate of water interception is decreased as 1/K is increased, particularly for values of 1/K larger than 1. An increase in Re0 also decreases the rate of water interception. The values of 1/K = 21 and Re0 = 187 represent the conditions for an airfoil with a 12-foot chord traveling at 400 miles per hour through a cloud composed of droplets 17 microns in diameter at an altitude of 10,000 feet. The corresponding value foryO'u- Y0,1 required for equation (5) is 0.022.

If the same airfoil is considered in flight at 200 miles per hour (droplet size and altitude not varied), the values of 1/K and Re0 change to 42 and 94, respectively, and the value of Yo ,u - YO,2 changes to 0.016, a decrease of 27 percent. The decrease in rate of water impingement (equation (5)) is 64 percent. The effect of speed on the rate of water impingement is large, because the spacing between the two tangent trajectories is affected by the speed and the speed appears directly in equation (5).

The variation of rate of water interception with airfoil speed is summarized for an altitude of 20,000 feet in figure 3, in which the is the total rate of water impingement per foot span of ordinate Wm/w airfoil per unit liquid-water content (g/cu m) in the cloud. The total rate of water impingement can be obtained as a product of the results in figure 3 and the liquid-water content existing in the cloud. Several chord lengths ranging in value from 2 feet to 20 feet have been considered in figure 3. The values in figure 3(a) are for flight through a uniform cloud composed of droplets 15 microns in diameter; and in figure 3(b), for 20 microns in diameter. As is shown herein, the effect of a change in altitude is a second-order variable; therefore, the results of fig- ure 3 are applicable over a wide range of altitudes.

Collection efficiency. - The collection efficiency of an airfoil has been defined (references 3 and 4) as the ratio of the amount of water intercepted to the amount of water originally contained in a volume of cloud swept out by the airfoil when at zero geometric angle of attack.

The collection efficiency - y0,j ) cos a Em T is presented in figure 4 as a function of the scale parameter . The preceding definition of collection efficiency permits a value of collec- tion efficiency greater than 1 for infinitely large droplets when the airfoil is at an angle of attack other than zero. The total rate of water interception can also be found in terms of the collection efficiency given in figure 4 for the 65i-212 airfoil from the relation

NACA RM E52B12 Wm 0.329 Em TLUw (6) where U is in miles per hour.

The value of the collection efficiency decreases with increasing values of the scale parameter r. At constant altitude, an increase in Jr is equivalent to an increase in the chord length with the same size droplets or equivalent to a decrease in the droplet size with the same airfoil chord length. The effect of wing taper can easily be obtained from the results of figure 4. For an airfoil with a 12-foot chord at the root section and a 3-foot chord at the tip section, the scale param- ii changes from 1675 at the root to 419 at the tip when the air- eter foil is moving at 400 miles per hour at an altitude of 30,000 feet through a cloud composed of droplets 17 microns in diameter (Re 0 = 90).

The collection efficiency increases from 0.19 at the root to 0.42 at the tip of the airfoil. Although the collection efficiency at the root is 19/42 as large as that at the tip, the amount of water impinging on the root section is 1.8 times the amount impinging on the tip section because the root section is four times as large as the tip section (equation (6)).

The collection efficiency for the NACA 651-212 airfoil is compared in figure 5 with the collection efficiencies of a Joukowski airfoil and an NACA 652- 015 airfoil, both of which are symmetrical and 15 percent thick, at an angle of attack of 4. The collection efficiencies for the Joukowski and NACA 652- 015 airfoils were obtained from reference 5. The 65i- 212 airfoil has a higher collection efficiency, in general, than the 15-percent-thick symmetrical airfoils, except for a portion of the curve at the lowest Reynolds number studied. The difference in collection efficiencies between the airfols of two thicknesses becomes greater for and large values of r, which correspond to the lower range of Em combinations of large values of chord length with small droplets. The higher collection efficiencies of the NACA 651-212 airfoil may be attrib- uted to the fact that the 651-212-airfoil is thinner and has a sharper leading edge than the other two airfoils and also has a slight camber.

A comparison of the collection efficiency between the 651-212 air- foil and the two symmetrical airfoils is also tabulated in table I for several conditions of speed, altitude, chord length, and droplet size.

With all three airfoils the effect on collection efficiency of a change in altitude from 10,000 to 30,000 feet is found to be small except for the conditions involving droplets larger than 40 microns in diameter.

The effect of altitude can be found from table I by a comparison of alternating rows.

NACA RM E52B12 The effect of altitude on the amount of water impingement can also be studied from the results presented in figure 2. The values of 1/K = 21.8 and Re0 = 325 represent the conditions for an airfoil with a 12-foot chord traveling at 400 miles per hour through a cloud composed of droplets 17 microns in diameter at an altitude of 1000 feet. An increase in altitude from 1000 feet to 30,000 feet decreases the values of Re0 to 92 and 1/K to 19.7 (speed., chord length, and droplet size changes from 0.021 to are not changed). The value ofyO'u - y0, 0.023, or an increase of 9 percent. The effect of moderate changes in altitude may be ignored, because the droplet size and the liquid-water content of the cloud are seldom known with accuracy sufficient to permit the rate of water collection to be calculated within 10 percent accuracy.

The effect of a change in droplet size on the collection efficiency is exemplified by a comparison of row 9 with row 11 in table I. A change in droplet size from a diameter of 50 microns to a diameter of 5 microns reduces the collection efficiency for the NACA 651-212 airfoil from 0.68 to 0.10. For this example, the altitude was assumed to be 10,000 feet; the flight speed, 400 miles per hour; and the chord length, 3 feet.

Extent of impingement. - A knowledge of the extent of impingement is necessary for the design of anti-icing equipment. The extent of impingement is determined by the point of tangency on the airfoil of the tangent trajectories. The farthest points of impingement on the upper surface are shown in figure 6(a); and those on the lower surface, in figure 6(b). The extents of impingement for the 651-212 airfoil are also listed in table I for some flight and operating conditions. The distances S and S 1 are measured on the surface from the point of intersection of the geometric chord line with the leading edge (fig. i) in terms of the chord length.

The extent of impingement along the upper and lower surfaces is summarized in figure 7 for the same speeds, chord lengths, droplet sizes, and altitude as given in figure 3. The extent of impingement on both the upper and lower surfaces increases with increasing speed and with decreasing chord length. It is much greater along the lower surface than along the upper surface. As an example, for an airfoil with a 12-foot chord traveling at 400 miles per hour through a cloud composed of droplets 17 microns in diameter at an altitude of 10,000 feet (Re 0 = 187 and 4, = 3590), the ratio of the extent of impingement to the chord length on the upper surface is 0.013 and on the lower surface is 0.09 (fig. 6).

Cumulative collection efficiency. - The cumulative collection effi- ciency for any trajectory intermediate between the upper and the lower tangent trajectories (fig. i) may be defined as

NACA EM E52B12 cos a.

- ( r0 - T The droplet impingement between the lower tangent trajectory and any other trajectory is shown in figure 8 as a ratio to the maximum impinge- ment. The ratio is YO - y0, E (8) Em - 70,u - The amount of water impinging on the airfoil between the farthest point of impingement on the lower surface and any other point of impinge- ment on the surface may be found from the curves of figure 8 and the relation W = 0.329 (L (9) EmTLUw cos a Em ) If a 12.5-foot-chord airfoil were traveling at 400 miles per hour at an altitude of 10,000 feet through a cloud composed of droplets 25 microns in diameter (1/K = 10, Re0 = 256, i = 2540), the ratio of the farthest point of impingement to the chord length on the lower surface is 0.14 (fig. 8(b)). The amount of water impinging between the 0.14 point and any other point, such as the 0.04 point on the lower surface, is found by substituting into equation (9) the value of E/E. for S = -0.04 and 1/K = 10 obtained from figure 8(b) ( E /Em = 0.24). The value of required in equation (9) is obtained from figure 4 Em = 0.23). For the airfoil in the preceding example, 24 percent of ( Em the total water impinges between the 0.14 and the 0.04 points on Wm the lower surface. For the same example, 84 percent of the total water Wm impinges on the lower surface of the airfoil and 16 percent, on the upper surface. In the limiting case of infinitely large droplets (1/K = 0), 65 percent of the total water impinges on the lower surface.

The effect of varying free-stream Reynolds number on the cumulative collection efficiency can be determined from figure 9. Figure 9 is also presented as an aid in the interpolation of results intermediate to the free-stream Reynolds numbers given.

Local rate of droplet impingement. - In the design of thermal anti- icing systems based on the principle of maintaining the water in the liquid state or on the principle of complete evaporation of the Impinging water, a knowledge of the local rates of water impingement is required.

The local rate of droplet impingement per unit area of airfoil surface can be determined from the expression

NACA RM E52B12 dy0 (10) WP = 0.329 Uw cos a = 0.329 Uw 13 cos a The values of 13 as a function of the airfoil distance S are given in figure 10. These values were obtained from the slopes of the curves in figure 8 by the relation 0,u 0,l In cyclical thermal de-icing systems a spanwise parting strip is usually located near the air stagnation line, which is located at S = 0.008 for the airfoil considered herein. The maximum rate of local impingement coincides very nearly with the proper location of the parting strip (fig. 10).

The data presented in figures 2 to 10 apply directly to flights in clouds composed of droplets that are all uniform in size. The water droplets in a cloud, however, are not necessarily uniform in size; the extent of impingement is always determined by the largest droplets in the cloud. The local rates of impingement are determined by the droplet- size distribution patterns present in the cloud. For flights in clouds composed of droplets that are not uniform in size, the curves of figure 10 must be altered to conform with the weighted basis of the droplet-size distribution.

CONCLUDING REMARKS Water-droplet trajectory data for an NACA 651-212 airfoil at an angle were calculated, and the collection efficiencies, impinge- of attack of ment areas, and rates of impingement were evolved from the calculations.

A graphical procedure is presented to aid in the interpretation of air- plane speed, chord length, altitude, and droplet diameter in terms of the dimensionless parameters used in the trajectory calculations. The rela- tively simple methods of solution will afford engineering data that were heretofore unavailable for the design of systems used in the protection of airfoils against ice formations.

Lewis Flight Propulsion Laboratory National Advisory Committee for Aeronautics Cleveland, Ohio

NACA RM E52B12 APPENDIX A SYMBOLS The following symbols are used in this report: droplet radius, ft a drag coefficient for droplets CD pressure coefficient C 6 ft) d. droplet diameter, microns (3.28x10 collection efficiency, based on maximum thickness of airfoil E p a2U inertia parameter, . w , where U is in ft/sec, dimensionless K L airfoil chord length, ft free-stream Mach number M absolute pressure, in. Hg p Remolds number with respect to droplet, 2a Pa -Re distance from an element of the vortex sheet to a point in the flow r field, ratio to chord length distance from chord line measured on surface of airfoil, ratio to S chord length maximum airfoil thickness, ratio to chord length T most probable icing temperature (fig. 14), OR Ta time, sec t flight speed or free-stream velocity, mph or ft/sec as noted U local air velocity, ratio to free-stream velocity u local droplet velocity, ratio to free-stream velocity v local vector difference between velocity of droplet and velocity, of V air, ft/sec

NACA.RM E52B12 rate of water Impingement, lb/(hr)(ft span) W local rate of water Impingement, lb/(br)(sq ft) in liquid-water content in cloud, g/cu w x,y rectangular coordinates, ratio to chord length CL angle of attack, 'deg impingement factor, dimensionless vortex strength, dimensionless r ratio of specific heats, 1.4 r viscosity, slugs/(ft)(sec) I coordinate points on airfoil, ratio to chord length , T density, slugs/cu ft P time scale tu/L, where U is in ft/sec, dimensionless 9L Re0 Pa '4!

scale parameter, a dimensionless Subscripts: free stream a air lower airfoil surface in maximum s airfoil surface upper airfoil surface u vortex v water w horizontal component x vertical component y

14 NACA RM E52B12 APPENDIX B METHOD USED TO CALCULATE INCOMPRESSIBLE FLOW FIELD AROUND AIRFOIL The velocity at the surface of an airfoil can be determined from a knOwledge of the pressure coefficient C and the free-stream Mach number M with the aid of the following expression: M : -. (cr + 2)11 us = + The pressure coefficients for a large number of points on the surface of the NACA 4 0 were obtained from 651- 212 airfoil at an angle of attack of wind-tunnel data taken at the Ames laboratory for several free-stream Mach numbers. The data are for a section of a wing having aspect ratio of 9. The surface velocities, which were used to calculate the flow field, were calculated for a Mach number of 0.2 for this report. The flow fields at other Mach numbers were not calculated becaue other exploratory work has shown that the effect of the compressibility of the air on the trajectories of the droplets was negligible.

The velocities in the two-dimensional flow field were calculated by distributing a sheet of vortices on the airfoil surface of such strength that the, velocities on the surface caused by the vortices were the same as the velocities calculated with the use of equation (Bi).. The velocity at a point in a flow field caused by an element of the vortex sheet of strength A placed a distance r from the point is 2tr If an element of vortex sheet of strength (u5 ES) 1th is placed on an increment LS of the airfoil at the position on the airfoil surface, the velocity caused only by the ith section of the airfoil is (us )i =

15 - NACA EM E52B12 at a point in the flow field at a distance rj from the ith section (fig. ii). The local components of the perturbation velocity, at a point in the flow field, caused by 300 vortex elements distributed on both the upper and the lower surfaces of the airfoil are v-' 2t r2 = 6 (B2)

=ajs

j=Q 2t r The horizontal and vertical components of the local velocity and UX respectively, are obtained by adding cos a to Uy, and sin a Ux,v to Uy,v.

A total of 300 vortex elements were used on the airfoil with a much denser distribution on the forward section than beyond the 50-percent point. Equations (B2) were solved with the use of electronic calculating machines. Approximately 300 points were computed in the flow field out to 1 chord length ahead of the airfoil in the region of interest with regard to computing the trajectories of droplets that strike the airfoil.

Between 1 chord length and 5 chord lengths ahead of the airfoil leading edge, the flow-field velocity components were approximated by assuming that the flow was caused by a single vortex located on the airfoil chord line 25 percent inward from the leading edge.

NACA EM E52B12 APPENDIX C GRAPHICAL PROCEDURE FOR INTERPRETATION OF PRACTICAL FLIGHT CONDITIONS IN TERMS OF DIMENSIONLESS PARAMETERS A graphical procedure is presented to aid in the interpretation of airplane speed, chord size, altitude, and droplet diameter into terms of the dimensionless parameters K, i, and Re0 used in this report. A solution of equation (2) is presented in figure 12 for two altitudes.

For given droplet diameters in microns and ratios of the chord length in feet to the flight speed in miles per hour, the reciprocal of the inertia parameter can be determined at altitudes of either 10,000 or 30,000 feet from figure 12. Altitude does not appreciably affect the value of 1/K, as can be noted from. either table I or a comparison of values in figure 12(a) with those in figure 12(b).

An airfoil with a 12-foot chord length at a flight speed of 400 miles per hour at an altitude of 30,000 feet passing through a cloud composed of droplets all of which are 17 microns in diameter will be used as an example in the graphical procedure to interpret practical flight units into terms of the dimensionless parameters. The value of 0.0300 1/K is obtained from figure 12(b) for the values of L == U 400 and droplet diameter of d = 17. The value of 1/K obtained from figure 12(b) is 19.4.

The free-stream Reynolds number for different altitudes may be obtained from figure 13. The product of the droplet diameter In microns and the flight speed in miles per hour must be known. The Reynolds num- ber is a function of the air density, which depends on the pressure and the temperature at the altitudes considered. The pressure used to cal- culate the air density was taken from tables of NACA standard atmospheric pressure at various altitudes, but the temperature was based on the most probable icing temperature at various altitudes. The most probable icing temperature was obtained from approximately 300 icing observations (reference 7) and is presented in figure 14. For the example under con- sideration, the product of the droplet diameter and the flight speed is (17)(400) = 6800. The value of . Re 0 as obtained from figure 13 is 89.

i4i for various values of 1/K and Re 0 may The scale parameter = 1732.

be obtained from figure 15. For the example considered, The following relations are presented for use when the degree of accuracy required is not attainable with the graphical procedure. The

NACA RN E52B12 17 values for the viscosity should be obtained from figure 16; these I.L values are based on the most probable icing temperature of figure 14.

K = 1.704X10- 12 _!

dPaU Re0 = 4.813X106 I.'

d. = 7.662X105 J = 2.826X106 PaT = 0.0412 Wm = 0.329 Em TLUw where d droplet diameter, microns collection efficiency (as given in fig. 4) Em K inertia parameter, dimenE\ionless L airfoil chord length, ft p absolute pressure, in. Hg Re0 free-stream Reynolds number with respect to droplet, dimensionless T maximum airfoil thickness, ratio to chord length most probable icing temperature (fig. 14), OR Ta U flight speed, mph maximum rate of water impingement, lb/(hr)(ft span) Wm w liquid-water content in cloud, g/cu m viscosity, slugs/(ft)(sec)

NACA RM E52B12 air density, slugs/cu ft Pa scale parameter, dimensionless (The density of water was assumed to be 62.46 lb/cu ft and the accelera- tion due to gravity, 32.17 ft/sec2.)

REFERENCES 1. Glauert, Muriel: A Method of Constructing the Paths of Raindrops of Different Diameters Moving in the Neighborhood 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 the Percentage of Drops Caught. R. & M. No. 2025, British A.B.C., 1940.

2. Langmuir, Irving, and Blodgett, Katherine B.: A Mathematical Inves- tigation of Water Droplet Trajectories. Tech. Rep. No. 5418, Air Materiel Command, AAF, Feb. 19, 1946. (Contract No. W-33-038-ac- 9151 with Gen. Elec. Co.)

3. Bergrun, Norman B.: A Method for Numerically Calculating the Area and Distribution of Water Impingement on the Leading Edge of an Airfoil in a Cloud. NACA TN 1397, 1947.

4. Guibert, A. G., Janssen, E., and Bobbins, W. M.: Determination of Rate, Area, and Distribution of Impingement of Waterdrops on Various Airfoils from Trajectories Obtained on the Differential Analyzer. NACA RN 9AO5, 1949.

5. Bergrun, Norman B.: An Empirical Method Permitting Rapid Determina- tion of the Area, Rate, and Distribution of Water-Drop Impingement of an Airfoil of Arbitrary Section at Subsonic Speeds. NACA TN 2476, 1951.

6. von Mises, Richard: Theory of Flight. McGraw-Hill Book Co., Inc., 1st ed., 1945.

7. Hacker, Paul T., and Dorsch, Robert G.: A Summary of Meteorological Conditions Associated with Aircraft Icing and a Proposed Method of Selecting Design Criterions for Ice-Prevention Equipment. NACA TN 2569, 1951.

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Figure 3. - Concluded. Rate of water impingement. NACA 651_212 airfoil; 40; angle of attack, altitude, 20,000 feet.

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NACA 651_212 airfoil; angle of attack, 4 0 . -

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