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UNITED STATES DEPARTMENT OF AGRICULTURE 

SOIL CONSERVATION SERVICE 
WASHINGTON, D. C. 



MEASUREMENTS OF THE FALL-VELOCITIES OF 
WATER-DROPS AND RAINDROPS 



By 

J. Otis Laws 
Assistant Soil Conservationist 



C. E. Ramser, Chief 
Hydrologic Division 
SCS-TP-45 
November, 1941 



K.TTED STAT2S DSPivRTMT OF aGRIC^JLTI'RE 
Soi] Conservation Service 
Washington, D. C. 



IffiASUREl^ENTS OF TI{E FALL-VELOCITIES OF 
WATER-DROPS AMD RA.INDROPS 



By 

J. Otis Lav.'s 
Assistant Soil Conservationist 



C. E. Ramser, Chief 
Hydrologic Division 



Contents 



Page 

INTRODTJCTION 1 

METHODS OF ISASIIREMT 

Camera and Light Box 3 

Magnif ication Error 5 

The Shutter 5 

■ ■ Taking the Picture 6 

Measuring Velocity from Negative 7 

?.^easuring -Diameter from the Negative 7 

Independent Measurement of Drop Size 9 

Unreported H'easurements of Very Small Drops . < . 10 
Adaptation of Apparatus to Measurement of Rain- 
drop Velocities 10 

Hygrometric Measurements 13 

TEST -RESULTS- • 

Indoor Measurements 14. 

■ 'MeasTirements of Raindrop Velocities 17 

Possible Sources of Error in the Present Meas- 
urements i . 17 

Height of Fall Required to Attain Terminal 

Velocity 19 

Explanation of Why Large Drops attain Terminal 

Velocities Sooner than Medium-size Drops .... 20 

Table 1. Test Data 23-27 

Table 2. Velocities of Falling Water-drops of 
Different Sizes After Various Heights 

of Fall . . . 32 

REFERENCES 33 



51Q5 



List 9f_ Illustrations and Diagrams 



■ Page 

Figure 1. ^Diagram of Raindrop" Camera 4- 

Figiare 2., Typical Velocity. Photograph , . 8 

Figure 3v .Diagram of Flour Calibration 12 

Figure /+. , Instantaneous Photograph of a Falling "Water-drop . 15 

Figure 5« Velocity of Fall of Seven Sizes of Water-drops after 
Heights of Fall of: 

a) From 0,$ to 3.0 Meters 28 

b) From 0,5 to 20.0 l^eters 29 

Figure 6, Diagram of Raindrop Velocity Data 30 

Figure 7. Smooth Curves Showing the Relation between Velocity 

and Diameter after T^.velve Different Heights of Fall 31 



MEASUREI.(ENTS OF TKS FALI-VELCCITY OF 
^vVATER-DROFS AMD RAINDROPS 
by 

J. Otis Laws 
INTRODUCTION ■ 

This paper presents measurements of the velocities of v.'ater- 
drops of sizes ranging from 1 to 6 mm in diameter, falling in still 
air from heights of 0.5 to 20 meters. A fe^/ measurements of rain- 
drop velocities are also reported. 

These measurements v/ero undertaken to assist in an under- 
standing of the action of rein, both real and artificial, in eroding 
soil. The drop sizt^s of rsins liave also been m.easured and will be 
reported separately. All of thtse studies wore carried out at the 
Hydraulic Labora.tory of th^- National Eijreau of Standards as a part 
of the work of the Soil Conservation Ser^'-ice. 

The velocity of falling raindrops and water-drops has been 
studied by several investigators in the past. The most complete mea- 
surements are those of P. Lenard (1) and Wilhelm Schmddt (2) (see 
"References" at end of paper) . 

Lenard measured the velocities of drops of 1.23 to 6.36 mm 
diam.eter by suspending them in the blast of a fan and measuring the 
air velocity'' at thcj point of si).spens:i on. Fe measured the size by 
catching thu drops on a sheet of absorbent paper as they slipped 
out of the air stream.. From the size of the wati^r spot he deter- 
mined the size of th.^- drop, following the method of drop size 



~ ^ - 

measurement used by most European observers. Defant (3) made the 
most thorough study of this method of drop size m.easurement . 

Schirddt measured the relocit^r of raindrops varying in size 
from. 0./+ to 3.5 nrni diameter with an ingenious apparatus consisting 
of tv;o discs mounted on an axle and turned at a known rate. A 
raindrop, i/vhich by chance fell through a sm^all sector cut in the 
upper disc, ".•ould land upon a niece of absorbent paper laid on the 
loT/\'er disc and turning v-zith it. The location of the spot relative 
to tlie projection of the sector on the paper gave a measure of the 
velocity, '.vhile the diameter of the spot gave a measure of the drop 
size . 

H. Fache (4-) made a fe'v mieasurenents photograohically but he 
had no v/sy of finding the size of the drops v/hose velocity he m.ea- 
sured. l^v'ache's ^/or^c deserves miention rrimarily because a modifica- 
tion of his method, vras used in the present measurements. 

The measurements reported here were undertaken primarily to 
obtain data on the velocities of drops falling through the rela- 
tively small distances that it is practicable to use in producing 
artificial rains. The data gc^thered on raindrops served as a 
direct check upon the m.easurum.ents of Schmidt. The indoor measure- 
ments at heights of fall where the velocity can be regarded as ter- 
minal offer a check upon Lenard ' s meas\irements . 

The measurements renorted here show a surprisingly large 
deviation from the measurements of Lenard and Schmidt, whose results 



- 3 - 

closely confirmed each other. Velocities fiilly 15 percent higher 
were observed. Finding no appreciable errors in the present mea- 
surements, it is the author's conclusion that the values observed 
nearly 30 years ago and accepted unquestioningly since, are in 
error by large amounts. 

l^THODS OF mS'IREJiENT 

Camera and Light Box; The measuring apparatus consisted of 
a 9 X 12 cm still camera miounted behind a "chopper disc", driven by 
a small sj^nchronous motor. In front of this a collim.ating lens was 
mounted at a distance equal to its focal length from the center of 
the camera lens. This had the effect of making objects on the other 
side of the collim.ating lens appear the same size regardless of 
their distance from the camera. Dark field illumination was accom- 
plished by a light box employing over run lamps arranged in such a 
way that light was directed toward the camera from, either side on 
the field of view. Strips of sheet metal were placed in front of 
the collim.ating lens to reduce the amount of light absorbed on its 
surface. An opening of A x 5 inches v^ras left in the light box for 
drops to fall through. The cam.era side of the collimating lens v/as 
shielded from, the light of the room by a conical casing. 

The use of dark field illumination and a "chopper disc" made 
it possible to obtain multiple imiages of a drop on a single film. 
The arrangements of the light and the cam.era are shown in the ac- 
companying schematic diagrairi. 



DROP VELOCITY CAMERA 



BACKGROUND PAINTED 
FLAT BLACK 



FIELD OF VIEW, 
4X5X7 INCHES 



-LIGHT BUNDS 





4 NO. I 

pMoto- 

FLOOD 
LAMPS 



COLLIMATING 
LENS 



O 
z 
m 

■n 
o 
Q 



•chopper' DISK/ 
MAKING 480 EX- 
POSURES PER SEC, 
DRIVEN BY A SYN- 
CHRONOUS MOTOR 



\ 1 


\ 1 


\l 








1 \ 




1 1 




In' 




1 ' ^ 





CAMERA 



1, The velocities of the drops were mens'ired by means of a still camera fitted 
with a rotating disc shutter driven by a synchronous motor and making ^80 
axposures per second. Dark field llluainatlon wns employed, and a colllmat- 
ing lens ^.s Interposed between the subject and the camera to eliminate mag- 
nification errors. 



Magnific ation Error: The magnification of the combined lenses 
was determined by photographing a millimeter ruled cross-section 
paper in the center of the camera field, the paper being held between 
two glass plates in a plane normal to the cam.era's axis. The accuracy 
of the rulings was checked just before photographing. Measurements 
from the developed film by means of a micrometer microscope showed 
the magnification in th6 center of the field to be 1/2.$^. 

The magnification at different positions in the field was de- 
termined by the same method with the following results: magnifica- 
tion 1/2 inch from the light side of the opening— 1/2.548, magnifi- 
cation 3 1/2 inches from the light side of the opening— 1/2 . 539 . 
The m.aximum error, therefore, to be expected from this source is 
about 0.2%, which was considered sufficiently accurate although the 
optical system was capable of even greater precision. If the colli- 
mating lens had not been used the deviation from the average magni- 
fication v:ould have been 8%. 

Mr. J. M. Slater (5) who suggested the use of such an optical 
system has described the optical characteristics of simdlar systems 
in a recent issue of Photo Technique . 

The Shutter: The timing device was a single phase 1/150 
horsepower s:(mchronous motor which turned at 30 r.p.s. on a 60 cycle 
alternating current. The motor was plugged into the lines of the 
Potomac Electric Pov^er Comrany. The generators of this company are 
tied in with the so-called "eastern system". The fruquency of this 
current is carefully regulated and believed to be constant within a 



- 6 - 

few tenths of a percent. To asceitain whether the motor, for any 
cause, was not tui'ning in synchronism v:ith the current when the 
"chopper disc" was mounted, the motor was placed in a dark room and 
iU-uminated by a neon lamp, under which illumination the screw heads 
of the hvib apuearud to be stationary, uinich would only occur vj-hen 
the motor '.ms turning in synchronism with the current. It vfas ob- 
served that coj^.siderable braking force had to be exerted before the 
motor "lost step" as indicated by an apriarent slovf counter rotation 
of the sci'ew heads. 

The "chopper disc" vas made from a paper chart designed for 
use on a vrater stage recorder. Ov.t of this circular sheet, which 
was 11 1/8 inches in diameter, 16 sectors viere cut 1/4- inch from 
the rim, each sector measurin,]; 1 x 7/S inches. Both sides of the 
sectors were carefully cut along the printed radii of the chart. 
The dj.sc was then paintedi black. 

Tak.in g tlie Picture ; To take a picture with this apparatus, 
the motor was started, the D.ights were turned on, and the canera 
shuttei' was opened for a short interval of time. It was desirable 
to have as many drops as possible on a single negative, but if the 
exposure was too prolong^.d the light absorbed on the irapt-rfectly 
shielded collimating lens obscur^;d the drop imag^. For this rea- 
son, an exposure of approxLmately 2 seconds was the m.aximuni that 
could be used . 

The aperture opening was set at f/Q. The film used was Agfa 
Super Pan Press. It v:as developed for maximum contract. 



Measurin g Velocity from Ne gative : The image of each drop ap- 
peared as a pair of broken lines, see figure 2. The partitions in 
the "chopper disc" obstructed the light many times during the drop's 
passage through the illuminated dark field. The two lines of the 
image were due to high lights which appeared on either side of the 
drop. 

To determine the velocity, the distance between the tips of 
two drop images was measured. The points of a pair of dividers were 
fitted to two images on the negative as far apart as possible; the 
"drop negative" was then replaced by a negative of the cross-section 
paper. The distance, h, as indicated by the points of the dividers 
was read to the closest half millimeter. The number, n, of breaks in 
the double 11 ne were then counted. Since the ruotor tu.rned at 30 r.p.s. 
and the "chopper disc" had 16 openings, the fi]jr; was exposed 30 x 16 ii 
4.80 times a second, or the time elapsed between the points of m.easure- 
ment is t = n/'4.80. Therefore, the average velocity over this interval 
is V = h/t = ^30 h/n. A correction for a slight error in the ruling 
of the cross-section paper photographed and used as standard of mea- 
surement was thrown into the coefficient, and the actual formula used 
in computing the velocity was v = 4. ,01 h/n. ViTien h is expressed in 
centLmeters this formula gives the velocity in m.eters rer second. 

Measuring Diameter from, the Negative ; The precise determina- 
tion of the diameter of the drop from tb.e negatiA'^e proved to be impos- 
sible. The high lights on either side of the drop produced an hala- 
tion on the filmi that varied with position in the field, velocity, and 
possibly, exposure, development, etc. Although several schemes of 



TYPICAL VELOCITY PHOTOGRAPH 




Figure 2. 

In the above typical photograph obtained with this camera the light blinds 
innediately in front of the collinjating lens appear as bri^t vertical streaks. 
Hie broken double lines were blurred imnges of falling drops, the drops haring 
flooved during the exposure, Eac^ drop produces a double line image due to the 
aaexiner in vdiich it was illTBdnated. 



>. 9 - 

measurement were tried, none was found that save consistent results 
within the rather narrow ILmits necessitated by this study. This un-^ 
expected short-coming of the ar^oaratus severely Ijjnited its utility, 
in so far as it neccssitattd tYnd use of independent methods of measur- 
ing the drop size. 

Independent ?>!"c:asur'vjm-ont of Drop Size i This measurem.ent rras 
accomplished by allowing water to drip from glass tubes of sm.all diam- 
'•:;tc'r, made by heating and drawing out glass tubing. Such a capillary 
vras fitted to a filtering flask v;hich was suspended above the raindrop 
camera and the flov/ throu;v;h the dripping tube v;as controlled by a scr^w 
clamp. Drops wort, allowed to ff.l] as fast as they could be conveniently 
counted— from two to three drops per second. Im,mediately before taking 
a photoe^raph, a weighing; bottle w.s placed under a capillary and 10 to 
30 drops were collected. T}iese were V'reiglied cn an analj/tical balance 
aiid the mass of the average drop was com.puted. Evaporation from the 
weighing bottD.e vms checked and corrections were applied when necessary. 

By this means drops rar)ging from 1.17 to 6.1 millimeters v^rere 
obtained. The production of drops smaller than 3 mdllimeters by this 
method necessitated the use of extremely small aperture,, and pressure 
often had to be applied to the filtering flask. To prevent the drops 
from v;etting the outside of the tube and to decrease the friction along 
the tube Y:alls, the inside and outside of these capillary tubes were 
coated with a thin layer of paraffin. This was accomplished by apply- 
ing a vacuum to the capillary and dipping it momentarily into molten 
oar af fin. 



Before this later technique was developed a number of measure- 
ments were made of drops of 2.3 millimeter diameter by forcing water 
through an uncoated capillary at a high rate of dripping — S to 15 
clrops per second. In order to determine the drop size under these 
conditions the rate of dripping was determined .by passing a ciece of 
blotting paper under the capillary for a given interval of tin^e and 
counting the number of spots. The weighing bottle vras then placed 
under the capillary for a measured interval of time. These m.easure- 
ments were not precise and the drops seemed to vary in size. A dis- 
proportionate variability appears in the data obtained by this method 
(see figure 3-b) . 

The usual procedure for measuring drop size did not give exact- 
ly the right value for the 6 mm drops, due to a secondary drop which 
is produced at the instant of breaking away from, the capillary and 
which fell into the weighing bottle along with the principal drops. 
Their size was roughly determined frorr measurements on the film, and 
a correction was applied although it amounted to less than 1%, 

Unreported Iv'easur ements of Ver^'" Small Drops ; Several measure- 
ments v/ere made of sprays coniposed of small drops but in these mea- 
surements no precise data were obtainable on their size. These mea- 
surements v:ere not considered suf f :.cient3y exact to report. 

Adaptation of Apparatus to ]\'."easurement of Raindrop Velocities ; 
In order to adapt this photographic apparatus to the measurement of 
velocities of raindrops, a means had to be provided for Treasuring the 
size of the individual drops photographed. A shelter was built on 



-li- 
the roof of the ^^ydraulic LaV.^oratory , and the carcera and its light 
box v/ere placed cJirectlv under a /i. x 5 inch opening in the roof of 
this shelter. The opening was insulated from splashes by grill Avork 
extending one foot out in all directions and was pro\''ided with a 
cover which could be removed by the operation of a lever extending 
inside the she].ter. 

The camera and the light box were prc^'ided with a broad belt 
with apertures so arranged that, by drawing this belt across the 
opening in the light box, the caj^.era field could be briefly exposed 
to the rain and, with the same movement, the film could be exposed. 
The apertures in the belt were so spaced that no drop could pass 
throu 'h the field without the film being exposed to it. a pan of un- 
compacted and calibrated flour v/as placed under the light box in a 
position to catch any drop that fel], through the camiera's field. Py 
drying and weighing the pellets of dough, form.ed by the drops in the 
flour, the size of the drops themselves could be calculated. See 
figure 3. (This technique of drop size measurem.ent was first used by 
Bentley (6) . The details of making such size measurement will be dis- 
cussed in a separate paper.) 

If the camera's field was not too long exposed to the rain, 
the few pellets found in the flour could be associated with the im.- 
pressions found on the negative by the size and location of each. 
Vi'hen too m.any drops fell through, the association of pellets and 
impressions became dubious and the filn... had to be discarded. 






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Successful measureiuents could only be made on fairly large 
raindrops falling nearlj- vertical. All drops aprroachin;^ the appa- 
ratus from angles greater than 30"^ fell outside the caj:iera's field. 
Single pellets of drops smaller than 2 or 3 it^'TC were difficult to 
weigh with sufficient precision. The chief limitation of this arrange- 
ment was the time required to obtain data on adequate range of sizes. 
I^Tiile many large drops were present in some of the rains photographed, 
vre were not fortunate enough to capture them in the extremely small 
samples peculiar to this apparatus. 

The vertical components of the velocities are reported in 
Fig\ire 6. 

HygroFietric H'easuremient s ; Hygrometric and barometric data 
were taken in conjunction vdth most of the indoor m.easurements . V/et 
and dry bulb temperatures were read from a sling osychrometer . The 
atmospheric pressure was determined by an aneroid barometer at the 
elevation of the caiaera. 



V 



I 



- V, - 



.. TEST RESULTS 

Indoor Measurem ents : The results of the indoor measurements 
are presented in table 1, where thaj are arranged in ascending order 
of drop size and height of fall. These data have been plotted in 
figures 5-a and 5-b . 

It was not always possible to obtain drops of exactly the same 
size. This necessitated drawing curves through data points that are 
not strictly comparable. The drop size represented by the curve is 
recorded along the curve. 

In the summarization of these data a theoretical frame-work 
would be most useful; its derivation, however, is difficult. The 
general problem, of which this ,study is merely a special case, is that 
of expressing the resistance to motion encountered by an immersed body. 
A concise review of the more important work clone on this problem from. 
Newton to present day investigators has been made by Rouse (7) in his 
"Nomogram for the Settli.ng Ve].ocity of Spheres." 

The deduction, of a formula applicable to water-drops is compli- 
cated by the change in shape that the larger drops undergo during 
their fall. A falling water-drop is not tear-shaped, as is now well 
known, but quite oblate at term.inal velocity as shown by the experi- 
ments of Lenard, loc. cit., as well as by some photographs of Pro- 
fessor Edgerton (12) of the Massachusetts Institute of Technology. 
I During the first few meters of fall a large drop vibrates between ver- 

tical and horizontal oblateness with a frequency that depends upon 
its size. This oscillation is shown, quite incidentally, in 



-15 - 




Figure k. This is a photograph of a large waterdrop falling 

throu^ still air. It was taken by H. E, "Bdgerton 
and confirms the observations of previous investigators vdio 
concluded that falling drops were flattened rather than "stream- 
lined" as commonly supposed. 



- 16 - 

Vvorthington' s (8) remarkable photo,iraphs . 

The frequency of the vibration of a water-drop was studied 
by Lord Rayleigh (9) who found U/T?- oscillations per second -where 
D is the diameter of the drop in- inches. 

The ef-f^ect of this change in shape is to add a new order of 
variation to a phenomenon already, beyond the reach of current ane.- . 
lytical tools. . If the shape of the drop had been .ascertainable 
from the photographs, it mi^iht have been possible to find an empiri- 
cal relation betvireen the coefficient .of 'the resistance form\ila and 
the Reynolds number. Th;e computation of this nwibor, hoYfever, re- 
quires a knowledge of the horizontal diajneter of the drop and, there- 
fore, the developm.ent of such an empirical formula, which would ha^re 
served as a basis 'for comparir:g the data obtained under the various 
conditions, was not possible. 

The original purpose of these measurements does not justify 
an extended rational investi:^ation, but it is hoped such a study may 
sometime be attempted. With a view of . assisting such an endeaAT-or, 
tlie hy,';^rometric and barometric data have been included in table 1. 
Liznar's (10) analysis of the data of Lenard and Schmidt is a ^ogd 
starting point for this study. 

Lacking satisfactory rational equations, the data have been 
very simply presented in graphical form. Figure 7 shows the relation 
between velocj.ty and dieoneter for A'arious heights of fall. These 
curves were obtained by reading off values from the smooth curves of 
figure 5. Table 2, in turn, has been prepared from figure 7. iit- 
tempts to extend tt.ble 2 to include smaller drop sizes encountered 



complexities similar to those discussed above, and it was thought 
best to confine the table to the range of experimentally deter- 
mined values . 

Meas urements of Rain drop Vel ocities ; The fev; observations 
gathered on the ^'-elocities of raindrops have simply been plotted in 
figure 6 and are not tabulated. A cuive representing the values of 
Lenard and Schmidt as well as the cun.'e representing the velocities 
observed indoors after 20 meters fall, have been plotted on the same 
figure of comparison. The values represent the vertical component 
of the actual raindrop velocities. 

The tendency for the velocities of raindrops falling from 
great heights to be lowei- than the velocities of water-drops fi.lling 
from only 20 meters indoors, may be due to the turbulence of the at- 
m.osphere through which the raindrops pass. The straj- points appear- 
ing in the figure are probablj^ the result of having incorrectly 
paired pellets and drop images. 

The velocities found by Lenard and Schmidt are evidently much 
too low. The low velociti.es obsen'-ed by Lenard may be due to exces- 
sive turbulence in the air stream he used. Schmddt's miethod of mea- 
suring drop size i.s open to question. In calibrating his absorbent 
parer, he aprears to have c^'erlooked the effect of height of fall upon 
the relation between spot diam.etcr and drop diameter. This oversight 
was not as important in Lenard' s measurements where the velocity of 
the drops relative to the paper was small. 

Possib le Sources of Error in the Present Measurements ; In 
attempting to explain the discrepancy between the velocity measurements 



presented here and tho£;e of Lenard and Schmidt, the technique of 
ineasureiTient einployod in this present study has been examined from 
many angles. Film shrinkage, motor speed and magnification were 
checked and no appreciable error "v;as found. 

Evaporation from the dron surface v^as also investigated. A 
few measurements of evaporation were attempted on largo drops fall 
ing 15 meters. The drop s:ize v;as measured at the dripping tube by 
the usual method, and inuned i.ately afterward . the drops were caught 
15 meters below in a tivo-quart jar in the bottom of which several 
inches of flour had been placed to cusicion the impact of the drops 
This jar Y^as capped and tlie drops v/ere again weig'hed. No evapora- 
tion was detected in tho-'-se crude measurements. 

A study of e-"-aporation by two Japanese is helpful in estima 
ti.ng the magnitude of this error. Nfimakawa and Takahashi (11) 
studied the evaporation of small water-drops in air and found the 
f o 1 1 ow i ng f o rmul a : 

dR = _/f . CM32 3^ _ ) 
dt R B ^ ^ . 

where R = radius of drop: t " time,* ''^ = density of dry air; f 
molecular diffusion constant; E - barometric pressure; pi - satur- 
ation pressure at the tcm.oorature of tlie evaporating surface; Po = 
vapor pressure at the temperature and humidity of the air; and-^ = 
a funt^tion of the velocity of the drop relative to the air. C.G.S 
units were employed. The value of was determined for drops of 
R - .085 eri'; foi- several velocities up to 2 m/sec. The results 



were expressed in the following empirical equations: 

= 1 + 0.21v'v' 

the velocity, v, being' expressed in meters per second. 

Pue to the lim.ited velocities used in their experiments a 
bold extrapolation must be made in appl;7ing these results to our 
problem. If such an extrapolation is justifiable, the Najnakawa 
formula indicates that the evaporation occurring in a fev; seconds 
of fall is negligible. The change in diairieter calculated for a 
1.7A inm drop after 2 seconds of fall was less than 0.1^. 

Height of Fall Re quired to Attain T erminal Velocity : The 
question is often asked, "Fow far must a drop fall to reach a con- 
stant velocity'-?" Curves dravm to fit the experimental data api^roach 
the terminal value gradually, and it is difficult to decide at pre- 
cisely what height they become coincident with the terminal value. 
The height required for drops to attain 95% of their tenninal veloc- 
ity can be found with greater exactness and, for most purposes, 
should serve just as well. These heights of fall, as derived from 
the curves of figure $, are: 

1 mm drop ■ 2.2 meters 

2 mm drop 5*0 meters 

3 rrm drop 7.2 meters 

4 nmi drop 7.B meters 

5,Fim. drop 7.6 meters 

6 mmi drop 7.2 meters 

It will be observed that the 5 and 6 mm drops attain 95% of 
their terminal velocity'- after shorter distances of fall than is re- 
quired by the 4 nim drop. Evidently, the necessary ht-dght of fall 
does not always increase with increasing drop size. This is also 
true when the heights are read at 9Q% of the terminal velocity and 



- 20 - 

the trend doubtless applies also to the height required to attain 
the full terminal velocity. 

Explanation of Why Large Drops Attain Tem inal Velocities 
Sooner than Medium-size Drops: This phenomenon is believed to be 
related to the apparent over-shocting of the terminal velocity 
observed in the 6.1 mm drop; see figure 5» It is too curious to 
be left without somu attempt being made at explanation. 

It will be observed that the 6.1 im drop appears to approach 
finally a terminr.l value of 9.30 m/sec. after about 20 meters fall. 
Drops of this size, however, were observed to be falLing at the 
same velocity after only 10.5 meters fall. In the intenrening dis- 
tance, a drop of this size apparently increases in velocity until 
it attains a maximum of 9.40 m/sec. at about the I4.th meter, after 
which height it gradually slows down. 

If this slovv'ing dovm is real — and it ap:oears to be — it is 
evident that the drop must have required time to attain its final' 
shape. For, if changes in shape occurred concurrently with changes 
in velocity, the final value of air resistance -would have been act- 
ing on a drop of this size after 10.5 meters fall, and this force 
of resistance being equal and ooposite to the force of gravity, 
viould reduce the resultant force to zero. The drop Y^ould then have 
continued in a state of uniform motion. 

- Since the velocity of the drop continued to increase, its 
shape must have been such that it offered less resistance at 10.5 
meters than it did at 20 meters, in spite of the fact that its 



velocity relative to the air was the same at both hei:;;hts. It 
nuivSt- be concluded, therefore, that changes in shape do not occur 
concurrently vn.th' changes in velocity, but that time i_s r equired 
^21! ^ r ela tive air velocity to pro duce the final drop sh ape 

correspondini^ t_o that velo city . 

Lenard's observations led him to a similar conclusion. He 
writes, "It can be seen that they (large drops svispended in upward 
air currents produced by a fan) are considerably deformed. The 
def onn.ation flattens the drop vertically; this often continues in 
the large droDS to a point whei'9 brcakifig-up occurs. Prior to 
this I have observed a similar deformation in natural rain by means 
of instantaneous j'.llumination. The drops of the previous test 
showed no sucli deformation. It appears, tVierefore, that this de- 
f orFiation requires more time thxan vvas allowed in the previous tests 
where less than a tenth of a second elapsed. This corroborates the 
fact that the flattening of large suspended drops, in which the de- 
formation occurs, begins only after an apiDreciable stay in the m.ov- 
ing air. 

"This time lag is understardable if the deformation is not 
the result of pressure directed perpendicular to the drop surface 
but of a tangential friction foi-ce of the air that brings the whole 
mass of the drop into circulatory m.otion which can onl^;- hapren 
slowly because of the Tel&t±velj large inei'tia of the y;ater." 

If such a lag ns present, the reason whj^ the large drops at- 
tain their terminal velociti.es at such small distances of fall is 



obvious. During the earl;,' stages of their fall, the large drops 
tend to roniain spherical and, thus cfff^.-ring a sr.aliler cross- 
sectional area bo the passing air, attain, ai'tei' small distances 
of fall, velocities higher than those consistent with their 
final shape. In the case of very large drops, the terminal veloc 
ity is actually exceeded. In drops of 5 ni^S the terminal veloc- 
ity is arnarently nuver exceeded, but it is attained very earlv. 
This tendency is also evident jn the i+ mm drop, although less 
striking:. The smallv^r eroi-^s are less easily defornied, and ap- 
Droach their terminal values of ■^^elocity more evenly. 



- 23 - 



TABLE 1 
Test Data 

D = diameter of a sphere equal in volume to the droD 
H = height of fall 

V = velocity of fall 

t = temperature of air (dry bulb) 
t'= wet bulb temperature 
B = barometric pressure 

V = kinematic viscosity of air 



D 

ram 


H 

meters 


V 

m/sec . 


t 
oc 


t' 

oc' 


B 

irtnHg 




(cm'^/sec. ) 


1.15-1.19 


0.53 


2.74 


21.0 


13.3 


752 


.153 


ti 


0.98 


3.48 


It 


II 


It 


II 


It . 


It 


3.47 


It 


It 


It 


It 


t» 


It .. 


3.44 


It 


It 


It 


It 


tt 


1.50 


3.87 


It 


II 


It 


II 


It 


2.50 


4.28 


It 


It 


It 


tt 


It . 


4.50 


4.60 


11 


It 


II 


11 


1.74 


0.50 


2.84 


23.0 


15.3 


748 


.156 


It 


It. 


2.85 


It 


It 


II 


11 


It 


1.50 


4.36 


*ti 


11 


It 


It 


It 


It 


4.38 


11 


It 


II 


It 


It . 


2.5 


5.12 


It 


It 


II 


fi 


II . 


3.46 


5.49 


It 


II 


II 


It 


1.73 


4.5 


5.69 


It 


It 


II 


.151 


tt 


6.0 


5.86 


II 


It 


II 


II 


II .. 


8.0 


5.90 


It 


It 


II 


It 


2.28 


0.5 


2.94 


25.0 


19.5 


764 


.155 


II ., 


II 


2.94 


It 


It 


tt 


ti 


It , 


1.0 


3.93 


It 


II 


It 


ti 


It . 


t! 


3.95 


It 


It 


It 


It 


2.29 


1.18 


4.33 


21.7 


10 .'6 


763 


.152 


ti 


II 


4.33 


It 


It 


It 


It 


It 


It 


4.31 


11 


It 


It 


It' 


ti 


It 


4.31 


It 


II 


II 


II 


2.28 


1.5 


4.67 


25.0 


19 .5 


764 


.155 


II 


It 


4.67 


It 


II 


It 


II 


2.29 . 


1.89 


5.10 


21.7 


10.6 


763 


.152 


tt 


11 


5.08 


11 


It 


It 


It 


II 


ti 


5.12 


II 


It 


II 


It 


tt 


ti 


5.12 


11 


It 


II 


II 


II 


It 


5.05 


It. 


It ' 


It 


It 




It . 


5.05 


It,. 


It 


It 


It 



1 



- 21, - 



TABLE 1 (Continued) 



J) H V 

rrini meters m/sec. 



2 .28 


2 ,0 


II 


II 


ti 


II 


ti 


2.5 


(1 


II 


2,38 


2.81 


It 


It 


It 


II 


2 .28 


3.0 


2.28 


■■' * J 


2.38 


3.71 


II 


II 


11 


II 


ir 


It 


II 


4.68 


II 


II 


It 


II 


It 


It 


2 .32 




II 


II 


II 


II 


2 . ' 


6.51 


II 


II 


11 


II 


11 


7.50 


II 




II 


10 8 


II 




It 


13.0 


II 


II 


II 


II 


3.02 


0.2/, 


II 


II 


II 


II 


II 


0.50 


It 


II 


It 


II 


II 


II 


3.08 


II 


II 


M 


II 


II 


II 


11 



5.15 
5.14 
5.13 
5.53 
5.49 
5.72 
5.80 
5.66 
5.83 
6.05 
6.21 
6,23 
6.23 
6.18 
6.36 
6,47 
6.50 
6.28 
6.66 
6.66 
6.64 
7.06 
7.03 
7.03 
6.92 
7.06 
7.06 
7.06 
7.07 
7.08 
7.08 

2.12 
2.12 
2,13 
2 -.96 
2.98 
3.01 
2.97 
3.04 
3.03 
3.04 
3.05 



t 

or 

25.0 
II 

II 

ti 

It 

21.7 

It 

It 

25.0 
II 

21,7 
It 

ti 

ti 

ti 

II 

II 

II 

24.0 
ti 

!l 
II 
II 
II 
II 
II 
It 
.tt 
II 
II 
II 



19.5 

It 

II 
II 
II 

■10.6 
II 

II • 

19,5 
II 

10.6 

II 

II 
II 
II 

It ■ 

II 

II 

15.9 

II • 

II 

II 

II 

II 

II 

II ■ 

II ■ 

II 

II 

It 

It 



_rijin_ H_£ 

764 
It 

It 

It 

II 

763 
It 

II 

764 
II 

763 
It 

II 

It 
II 
II 
II 
II 

761 
II 

II ■ 

I! ■ • 

II • 

II 

It 

II 

II 

II 

II 

II 

II 



(cm^/sec . ) 

.155 

II 

II 
ti 
II 

,152 

It 

II 

.155 

II 

.152 
It 

II 

It 

It 

II 

It 

II 

.154 
It 

II 

II 

II 

II 

II 

II 

It 

II 

II 

II 

It 



23.6 
II 

tt 

1! 



15.0 756 

It II 

II II 
II II 



,151 



- 25 - 



TABLE 1 (Continued) 



U 


IT 

n 






f 1 


p 

D 


V 


irjn 


iTi T* /~^ "v^ O 


iTi/ S C . 


UU 




luTu ng 


cm / ssc • ) 






/ no 










It 


If . 


/ '0'^ 










ft 


it 


/..■07 












"1 n 




<:J .0 


1.1 \ U 


/50 


. 15-5 


II 


It'- 


4 . -LJ> 




II 


II 


It 


II 


It 


4 1 ^-^ 


It 


II 


11 


It 


II 


It 


4 ■ .^ . J 


It 


It . 


11 


It 




X • ^ 


^4- . J O 


9^ 


91 Q 






ti 


1 s ^^'.J 




tt 


tt 










''O 


II • 


II . 






II 


It 


^ ,' ^ 


II 


It 






II. 


II 


c ' f, 

^ • •■: 


II 


It 






II 


2 ^0 




II 


II 






11 


It 


^. , '-0 


II • 


11 






11.. 


^ % u J.. 


^ ■ -«. O 


II 


It 






It., 


II 


O - . .if 


II ■ 


11 






It 




/ . ^ 

U • ..J 


II 


II 






li , 


It 


D •4.; 


II 


II • 






II , 




/ '' • ' 

u O'X 


II 


II 






It. , 


It 




tt ' 


It 








/ 09 


U .■ /if 


21 








II 


II • 


A HQ 


II 




II 




It 


II 


A no 
U /V 


It 




11 




II . 


/ ';o 


/ < UU 


on n 
>cU « / 




/44 




It . 


II 




It 




It 




It 


II 


/ c UU 


II 




tt 




II 


5 • U4 


I • ±0 


oi 
<;x .U 




n c c 

I J J 




II 


It 


n 1 A 
/ . 10 


II 




It 








7.16 


If 




f 1 




It 


5 •4y 


/ . (L<. 


It 




tl 




It 


o . uu 




ti 




It 




It 


II 




It 




It 




II 




n 1 

/ "47 


II 




tt 




It 


It 


/ 51 


It 




II 




11 


/ .UU 


^"7 r n 

/ -. / 


11 




II 




II 


7.50 


7 . ^..'0 


20.7 








11 


8.00 


7-79 


It 




tl 




It 


II 


7-82 


II 




II 




It 


tt 


7.82 


II 




II 




3.00 


10.00 • 


7 ■ 90 


24.8 


21.2 


• 756 '. 


.156 


II 


It 


7.82 


It 


It 


11 




II 


12.00 


7.94 ■ 


II 


It 


It 




II 


II 


7.96 


11 


tt 


II 




II 


14.00 


3.08 ■ 


II 


It 


tt 




II 


II 




II 


It 


tt 





- 26 - . 

TABLE 1 (Continued) 



D 


H 


V 


t 


t' 


B 


V 


Km. 


meters 


m/sec . 






mm Hg 


( cm.'^/se 


3.00 


16.00 


8.09 


24 » 8 


21.2 


756 


.156 


11 


11 


8.15 ■ 


It 


II 


II 


II 


II 


18.04 


.,8.11 


It 


It 


It 


II 


II 


II 


8.11 


It 


11 


It 


tt 


1! 


20.0 


8.11 


11 


11 


II 


11 


3.92 


0.88 


3.97 


25 .0 


19.0 


762 


.155 


II 


II 


3.96 


II 


1! 




11 


II 


1.36 


/ Ol 


11 


11 




11 


II 


II 




II 


11 




II 


It 


1.88 


5.-^7 


11 


II 




11 


II 


2.38 


6.04 


M 


II 




II 


II 


II 


6.02 


11 


II 


" 


II 


II 


2.88 


6.43 


II 


II 




II 


II 


II 


6.35 


II 


II 




It 


II 


3.39 


6.81 


It 


n 




It 


II 


6.00 


8.00 


21.6 


16.1 


750 


.154 


II 


11 


8.04 


II 


tt 




It 


11 


9.00^ 


8.65 


II 


II 




II 


II 


II 


8.62 


If 


11 




It • 


11 


11 


8.58 


11 


II 




It 


It 


12.00 


8.73 


11 


11 




!I 


II 


II 


8.71 


11 


11 




tt 


It 


II 


8.83 


II 


It 




ft 


It 


It 


8.64 


11 


It 




It 


II 


It 


8.70 


II 


II 


„ 


It 


It 


15.00 


8.78 


It 


II 




It 


It 


II 


8.77 


II 


II 




11 


It 


II 


8.77 


It 


fi 




II 


It 


18.00 


8.87 


11 


II 




It 


It 


II 


8.84 


II 


It 


" 


11 


It 


20.00 


8.88 


II 


It 


"- 


11 


II 


II 


8.84 


11 


It 




It 


11 


II 


8.82 


. It 


It 




11 


II 


II 


8. 88 


II 


It 




It 


5.0/^ 


1.0 




25.0 


19.0 


762 


.155 


II 


1.5 


5.16 


It 


11 




II 


It 


tt 


5.13 


It 


I! 




II 


II 


II 


5.14 


It 


11 




II 


II 


It 


5.13 


It 


II 




It 


II 


2.0 


5.82 


It 


It 




II 


It 


2.5 


6^40 


■ 11 


It 




It 


It 


II 


6.35 


11 


II 




II 


It 


3.0 


6.85 


It 


It 




tt 


It 


It 


6.92 


1! 


II 




It 



- 27 - 



TABLE 1 (Continued) 





u 


V 


t 


tt 


D 






rripf pT'C 


ill/ WWW • 


OQ 


oc 


mm W(f 


( CTCl'~ /KPf 

\ Wilt / 




3.A6 


7.22 


25 .0 


19.0 


762 


.155 


It 


II 


7.17 


II 


It 


II 


It 


nn 


6 nn 




21 6 


lA 1 

XW • X 


7S0 

/ >w 


.15A 


II 


n 





It 




II 




ti 


II 




II 


„ 


It 




II 


Q 00 


P .'^6 

• / W 


II 


„ 


It 




II 


II 


8.96 


II 




It 


„ 


It 


II 


8.96 


II 


„ 


It 




II 


15.00 


9.07 


II 




II 




II 


M 


9.28 


II 




It 




» 


IS .00 


9 .20 


II 




tl 




It 


II 


9.20 


II 




It 


n 


It 


It 


9.20 


II 




It 


M 


It 


It 


9 .20 


II 


„ 


11 


„ 


It 


It 


9.35 


II 




It 




It 


II 


Q.20 

J • ^w 


II 


„ 


It 


„ 


It 


20 00 

^W • WW 


Q 28 


II 




tl 


„ 


II 


II 


9.25 


tt 




it 


„ 


It 


11 


9.28 


II 




It 


„ 




1 00 

J~ •\J\J 


A. 33 


24.0 




762 

/ w^ 




It 


1 '^O 

A. • ^W 


2/ 


tl 




II 




H 


II 


.20 

J • ^W 


II 




II 




tt 


2 2^^ 


6 23 

W » 


It 




II 




11 


II 


A 20 

\J • *cw 


It 




tl 




It 


00 


6. PS 

w • ^ 


II 




It 




II 


II 


6 86 

W • W W 


II 




II 




(J • X.\J 


A on 


SP 


21 A 


1/ A 


7S0 

1 :?w 


1 5/ 


II 


Q on 

7 . WW 


Q 1 S 
7 . -LP 


It 


II 


It 


II 


tl 


1 2 nn 




II 


11 


It 


II 


It 






II 


It 


II 


II 


(r-.YL 


16.00 

_l_ W- •WW 


Q . ^^ 3 


It 


It 


It 


It 


6.05 


17.20 


9.15 


22 . c 


20.3 


764 


.153 


It 




9.12 


tl 


It 


It 


II 


It 




9.21 


It 


11 


It 


II 


11 




9.32 


It 


11 


It 


II 


It 




9.17 


II 


11 


It 


11 


11 




9.21 


It 


II 


It 


II 




18.0 


9.30 


21.6 


l/,.6 


750 


.154 


6.05 


19.7 


9.17 


22.8 


20.3 


764 


.153 


It 


It 


9.23 


It 


II 


II 


II 




19.85 


9.35 


21.6 


1A.6 


750 


.154 




/ 0.5 1.0 1.5 2.0 2.5 3.0 

/ HEIGHT OF fall: METERS 



Figure 5a.- velocity of fall of 7 sizes of woterdrops after heights of fall of from 0.5 to 3.0 meters. 



9-17-40 N-3247-1 



1 



- 30 - 



o 
z 
o 
o 
III 
</) 

QC 
UJ 

a 

</) 
q: 

UJ 

»- 

UJ 



o 
o 















































1 Indo 


or Ma 


oture 


m«nt« 




















afttr 20 r 
of fall 


neters height 
















♦ 




- — ' 
















— 








^Acc< 

and 


jrding 

Schml 


to L< 
dt 


inard 










_J 


▲ 
























/ 


















































1 


























V 


























/ 






• 

▼ Roin of March 27, 1939 
X Rain of April 1 ,1939 
■ Roin of Moy 22,1939 

• Roin of Junt 13, 1939 
A Roin of June 30, 1939 

♦ Rain of July 26, 1939 


























J- 
















1 








































/ 









































































































3 4 

DIAMETER: mm. 



6 



Figure 6.- Raindrop velocity data, together with curvet showing the results of 
Lenard ond Schmidt, and the results obtained with waterdrops after 20 me- 
ters foil. 



I 



9-17-40 N-3247-3 



1 




3 4 
DIAMETER : mm. 

Figure 7 - Smooth curves showing the relation between drop diometer ond velocity after 
12 different heights of foil. 



9-17-40 N- 3247-4 



- 32 - 



TABLE 2 



Velocities of Falling "feter-drops of Dif-f'erent 
Sizes After Various Heirrhts of Fall 

Velocity in meters per second; height of fall, H, in meters 
diameter, D, in pil? imeters 



0.5 0,75 1.0 



.5 2.0 2.5 3.0 4.0 5.0 6.0 8.0 20 



1.25 


2.65 


^ . J- ^ 


1.50 


2.76 


3.26 


1.75 


2.34 


3.34 


2.00 


2.89 


3.40 


2.25 


2.93 




2.50 


2.96 


.^.50 


2.75 


2.98 


3.54 


3.00 


3.00 


3.58 


3.25 


-^.02 


^.61 


3.50 


3.03 


3.64 


3.75 


3.04 


3.66 


4.00 


3.05 


3.67 


4.50 


."^.07 


3.70 


5.00 


3.09 


3.72 


5.50 


3.10 


3.74 


6,00 


3.10 


nc 



3.52 
3.64 

"^.74 

O JO 

"^.98 
4.04 
4.09 
4.12 
4.15 
4.18 
4.21 
4.24 
4.27 
4.29 
4 c 31 



3.97 


4.21 


4.43 


4.56 


4.80 


4.85 


4.85 


4.85 


4.85 


4.18 


4.50 


4.82 


4.99 


5.25 


5.39 


5.47 


5.51 


5.51 


4.34 


4.73 


5.10 


5 . 31 


5.64 


5.80 


5.92 


6.08 


6.08 


4.47 


4.92 


5.29 


5.55 


5.91 


6.15 


6.30 


6.53 


6.58 


4.57 


5 . 07 


5.44 


5.74 


6.14 


6.42 


6.63 


6.90 


7.02 


4.65 


^^.19 


'".57 


5.89 


6.^4 


6,67 


6.92 


7.22 


7.41 


■M- . 72 


5.23 


5.69 


6.02 


6.52 


6.89 


7.16 


7.50 


7.76 


4.79 


5.37 


5.80 


6.14 


6.68 


7.08 


7.37 


7.75 


8.06 


4.35 


5.45 


^^.89 


6.25 


6.82 


7.25 


7.56 


7.96 


8.31 


4.90 


5.52 


5.93 


6.35 


6.95 


7.40 


7.73 


8.15 


8.52 


4.95 


5.58 


6.06 


6.44 


7.07 


7.53 


7.88 


8.31 


8.71 


4.98 


5.63 


6.12 


6.52 


7.17 


7.65 


8.00 


8.46 


8.86 


5.05 


5.72 


6.24 


6.66 


7.36 


7.85 


8.21 


8.70 


9.10 


5.11 


5.79 


6.33 


6.77 


7.50 


8.00 


8.36 


8.86 


9.25 


^.16 


5o85 


6.40 


6.86 


7.61 


8.11 


8.47 


8.97 


9.'^0 


5.20 


5.90 


6„46 


6o94 


7.69 


8.20 


8.55 


9.01 


9.30 



- 33 - 



REFERENCES 



(1) P. LENARD, "Ueber Regen," Meteorologische Zeitschrift, Vol. 21, 

pp. 248-262, 1904. An English translation of this 
article may be obtained upon application to the 
Librarian, Soil Conservation Service. 

(2) l/Y, SCMIDT, "Eine iinmittelbare Bestimmung der Fallgeschwindigkeit 

von Regentropf en," Sitzungsberichte der Mathernatisch- 
Naturwissenschaf tlichen Klasse der Kaiserlichen Akademie 
der Wissenschaf ten, CXVIII Band, Abteilung Ila, Vvien, 
1909* An English translation of this article may be 
obtained upon application to the Librarian, Soil 
Conservation Service. 



(3) A. DEFANT, "Gesetzraaessigkeiten in der Verteilung der verschiedenen 

Tropf engroessen bei Regenfaellen, " Sitzungsberichte der 
Mathematisch-Naturwissanschaf tlichen Klasse der Kaiser- 
lichen Akademie der liiissenschaf ten, CXIV Band, Abteilung 
Ila, pp. 535-646, 1905. 

(4) H. LLACHE, "Ueber die Geschwindigkeit un Grosse der Regentropf en, " 

Ifeteorologische Zeitschrift, Vol. 39, p. 278, I9O4. 

(5) J. M. SLATER, "Perspective Control in Close Up Photography," 

Photo Technique, iJovember 1939. 

(6) W. A. BENTLEY, "Studies of Raindrops and Raindrop Phenomena,"' 

Monthly "yYeather Review, pp. 450-456, October I904. 

(7) H. ROUSE, "Nomogram for the Settling Velocity of Spheres," 

Exhibit D of the Report of the Committee on Sedimen- 
tation, 1936-1937, pp. 57-64, National Research 
Council, Division of Geology and Geography. 

(8) A. M. WORTHINGTON, "A Study of Splashes," Longmans, Green & Co. 

(9) LORD RAYLEIGH, "On the Capillary Phenomena of Jets," Proceedings 

of the Royal Society, XXIX, pp. 71-97, 1879. 

(10) J. LIZNAR, "Die Fallgeschwindigkeit der Regentropf en," 

Meteorologische Zeitschrift, Vol. 3I, pp. 339-347, 1914. 
A translation of this article may be obtained by appli- 
cation to the Librarian, Soil Conservation Service. 

(11) T. NA:i..IEKAM and T. TAKAHASHI, "Note on the Rate of Evaporation of 

Small Water-drops, such as Raindrops," Memoirs of the 
College of Science, Kyoto Imperial University, 
Series A., Vol. XX, No. 4, July 1937. 

(12) E. K. EDGERTON and J. R. KILLIAN, "Flash: Seeing the Unseen by 

Ultra-High-Speed Photography," Hale Publishing Company, 1939.