ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR DETERMINATION OF DROP TRAJECTORIES BY MEANS OF AN EXTENSION OF STOKES' LAW By P. SHERMAN Jo S. KLEIN M. TRIBUS Project M992-D AIR RESEARCH AND DEVELOPMENT COMMAND, USAF CONTRACT AF 18(600)-51 April, 1952

SUMMARY Trajectory data were determined for drops in air flowing over a cylinder, a sphere, a ribbon, and several airfoils, by reduction of the number of parameters previously used. One trajectory data curve for each body was thus obtained where an entire family of curves was previously necessary. This reduction of the number of parameters was suggested by Langmuir. ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN DETERMINATION OF DROP TRAJECTORIES BY MEANS OF AN EXTENSION OF STOKES' LAW The equations for two-dimensional motion of spherical drops in flowing air (considered incompressible) may be written in non-dimensional form as follows: dvx CDR Kvx (U - vX) (1) dx 24 dv CR K y (u - v (2) Y dy 24 Y Y dx - dy v = - = -- x dt ' dt where2aUpd S (4 9L A( L \ is the distance the drop will travel if projected into still air with velocity U(in the absence of gravity) if the drag force obeys Stokes' Law. Langmuir and Blodgettl found that results obtained by the differential analyzer solution of (1) and (2) were in good agreement with values resulting from calculations based on K', an adjusted value of K, and Stokes' Law, where (for flow around a cylinderS: K] - 1/8 = (K - 1/8). (5) O h~~~~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN X is the distance a droplet will travel if projected into still air with an initial velocity of U when the drag coefficient follows values given in reference 1. \/ks is obtained from: RU. 1 24 (6) R CDR s R \U d (6) 0 X,/% is therefore an average value of CDR/24 for a drop projected into still air with an initial velocity U and final velocity zero. If, for the case of still air, Eqs. (l)and(2)are solved by using an average value of CDR/24EK, there is a K for which we can apply Stokes' Law and obtain the same trajectory. Ko is defined by: /1 C]R 1, (7) K 24average Ko or: K = K. (8) 0 o;s If this adjustment of K is extended to the case of a varying airvelocity field, the curves of Figures 1-20 result. An examination of the curves suggests that for each body, instead of a wide spread of curves as shown in the Guibert2 and Langmuir and Blodgett reports, only one curve is then necessary. ___________________________ 2 ____________________________

.9 ~~~.8~,o:loo 7 --- =1000 w.6:0ooo 0 iL..5 I/Z.,^~~~~~ ~CYLINDER EM vs. Ko / /.1 -.1.2.4.6.8 1.0 2 4 6 8 10 20 40 60 80 100 Ko FIGURE I

900 L) 800 0 z 7011 " =0 o =,oo O 600 =104 ~~~0~~~~~~~~~ ~~ o /' I --- — Lz ~ =105 50~ 0 400 -j CYLINDER z / <,' 20~s~~~~~~~~~~~~~~~M /vs K.1.2.4.6.8 1.0 2 4 6 8 10 20 40 60 80 100 Ko FIGURE 2

.97_ ___ __ 4 — 100.S/ I K II6.6 1 =- o.1.2.4.6.8 1.0 2 6 8 10 20 40 60 80 100 Ko FIGURE 3 7I....1.2.4.6.81.0 2 4 6 810 20 40 6080100~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i /.:7/11: -- iCYLINDE! ~j?;/, ~ ~ ~ iGR 3,:~~

.9 ----.8,.7/- R= 64 Ru = 16 z /'// - RU =256 | 6 --------- -- - --- -------- ^ - - --- - - - -- - -=1024 ILL - -NACA 652-015 AIRFOIL SYMMETRICAL 15% THICK.2 = EM vs K0.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 10 Ko FIGURE 4

w OL 0 (C) crw a. 3I --- I RU =64 z w RU =16 w 2 -3 RU-256 ~C3 -— ~Ru=1024 z LL o z 20 W w o iE ^/C- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~0 NACA 652-015 AIRFOIL cr. ---__ SYMMETRICAL 15% THICK ool"Q a=40 Ct). ---^ ~ ~~~SU vs Ko.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 10 KO FIGURE 5

0 cn~.401 1 I I I i 1 I i i I ~Y~ 4t — ~R 64 w z.30 Ru 16 _ o- Ru =1024_ I uI 0 Z. 20 o- NACA 65-015 AIRFOIL I U)^~~~~~~~~~~~,SYMMETRICAL 15% THICK - / _~ a:I Ct~~~~~~~~~~~~~~~~~~~ 40 -.10 IL %- ^ I` i i i; I I I / I i SL vs Ko.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 10 Ko FIGURE 6

|\- R 512 oIJ ' *// '' I\ I\ '. s128 E I i i t I J ^ t I I JOUKOWS IRUI 8.5 Lu JOUKOWSKI AIRFOIL SYMMETRICAL 15% '000 ^ ^ / " ' ' \ \ \, ja = 0 0.01 02.04.06.08.1.2 4.6 8 1.0 2 4 6 8 10 Ko FIGURE 7

2.40 w z a. o -- Ru =2048 - R =512 L.30- --- Ru=128 0~ __ --- Ru 32 z. 0 Ru - 8.20 LL J OUKOWSKI AIRFOIL __ '_j *../..- I SYMMETRICAL 15 % THICK -;' -i ~ o ~ a! =0~.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 10 K0 FIGURE 8

I II I I I I I I I I YI5.8.7 - - - - - Ru i 64 >- I I I I I I I I i I I I I I I I I I RU = 16 - R u = 256 L.. 6 - - - ------ RRu =1024 0.5 I~~~~-J~~~~~~~~~~~o.3_ ___________I III/JOUKOWSKI AIRFOIL _______/ ____ - - - - - -SYMMETRICAL 15% THICK /a=20 KO FIGURE 9

w U') Q: W0A O. w Ru: 16 _z- Ru = 256 a..30R --- —-- - - - - - ----- - - _ _ u - = 1024 0.4. 0 Q. 0 I: LL. 0 u'0 u ^ JOUKOWSKI AIRFOIL ___ __ _ 00,,__. SYMMETRICAL 15% THICK a.10 - U '.' a-20.01.02.04.06.08.1.2.4.6.81.0 2 4 6 810 KO FIGURE 10

CU) w:L Ru;664 Ru =16 aL.30 - Ru = R256 - 0 -L1 g, ----- --— ____ __-_-_-__-____- -- SYMMETRICAL 15%__ THICK... - z.a=20.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 810O 0 JOUKOWSKI AIRFOIL w _________ __ _ _ ^g^^ _______ __ _ _ ___ SYMMETRICAL 15% THICK ~01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 60 KO FIGURE II

.9.7 ------ - _ _ ----- _ _ - _- - Ru =64 - >^ --- RU =16 z RU = 256.6 1024 ~~~~u~~~~~~ ~~~~- ---- -- 12 LL. U. J I-.4.3 JOUKOWSKI AIRFOIL 2 _____ _ __ _ _ _ __ _ __ __ _ _ ___ SYMMETRICAL 15% THICK a=40.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 10 Ko FIGURE 12

UJ 0 () (.40 Q: I -_ z bJ _ 2 __ _ Ru _64 wS --- RU 16 z -- Ru =256 ^.30 - --- Ru = 1024 QI 0 -0 oci ______ ___ ____ __ _.___ _ ix IL C3 0 z.2C FA 0 riI.IC I-.^ ^ - at% t^ ^ I I I I / I III JOUKOWSKI AIRFOIL IL _______ _ __^ S __ __ SYMMETRICAL 15% THICK o? ^-^^S^'^^ a =40.01.02.04.06.08.1 2.4.6.8 1.0 2 4 6 8 10 Ko FIGURE 13

w 0 ILcrr _____ w.40 --- R~ 64 Ru = 16._ -- Ru = 256 ------ - --- Ru = 1024 Z.30 0.20 0 0, a. JOUKOWSKI AIRFOIL __-, __'___SYMMETRICAL 15% THICK f).10 ----- I' l a a40 I. -J Cl).01.02.04.06.08.1.2.4.6.8 I. 2 4 6 8 10 KO FIGURE 14

.8_ —.7 ------- ----- - - Ru 64 > I I I I I Ru:= 16 ______ / / // I --- Ru: 256 ---- - - _ --- —.6 --- - - RU = 1024 ULL /L 1.5 0 I"-.^/~/ JOUKOWSKI AIRFOIL CAMBERED 15% THICK a =o~.01.02.04.06.08.1.2.4.6.8 1.0 2 4 6 8 10 KO FI'GBJRE 15

LU i - I0J < LL i --- Ru = 16 I... --- RU=256 (3 ___ __ -|024 - - 4 0 aL. 0.2o 0 0. 03.10 w 3 I:^ JOUKOWSKI AIRFOIL Icfr ^^^CAMBERED 15% THICK a. a=o0 (n.01.02.04.06.08.1.2.4.6 8 1.0 2 4 6 8 10 KO FIGURE 16

LLI 0 F ~.4C I- -, z w- RU= 16 (. LLD/ 1 I I Ij j --- = I T T i X( - 1- — RU=256 0.30 Ru 1024 z a. 0 QM:) LLI 0O~~~~~~~~~~K FI R 10 ~~~C,,L ~ ~ ~ ~ ~ ~ ~E, K0

.9 o / —100.6 / L-.6 _ 4 6 LK.3/ SPHERE.01 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 20 40 60 80 100 KO FIGURE 18

OM ANGLE BEYOND WHICH NO DEPOSITION OCCURS _ N 0 a n C) 0 O Q 0 __ -. ~~o ~o ~ o.... A b b *^ -----------------------. --- —----.,._....... _.____..... - M 0 po 0 - --- - - o0 4_. -,__ -_ '_,~1 l l II IX II.IoO o~ --- -- Q -------------------------------------- 5 ---- ' --- —------------- ""~-~~~~~~.....

.8 --------------.7 --------- --- -__- TI-_- ----— r - -- -______ ---, --- - _ f -100 z6 - I i i I i I// i i I I i -- f __ _1000 LaJi~~/ / 1/ l u ' I// I.8 U2 w//// 7 ~~~~~RIBBON.1.2.4.6.8 1.0 2 4 6 8 10 20 40 60 80 00 Ko FIGURE 20

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The curves of Figures 1-20 were computed in the following manner: the value of RU under consideration. Finally Ko was obtained from the formula K = K A/A. This results in a table of values giving EM as a functio n of Ko. Several values of R were chosen for each body so that a family of curves of EM versus K with RU as parameter results. The curves pertaining to the sphere, cylinder, and ribbon were obtained in the following manner A value for was hosen for a paberticular body and several values of EM ('or 9M or V1, as the case may be) and corresponding values of K were taken from the Langmuir and Blodgett curves. From R = (K)l/2 an RU was calculated for the chosen value of 0 and each value of K. The ratio X/A was then obtained from Table I of the Langmuir and Blodgett report. Ko was then calculated from K o = K /s and EM (or ng or V1) as a function of Ko was plotted. This procedure was repeated ffor several values of 0 for each body. The following curves are examples of those used in he above calculations. _________________ 25 _____calculati

EM MAX 104.6 -^ - ------- ^NACA 652-015 AIRFOIL 5 on ^vV ^s L^S^^ 4 SYMMETRICAL 15% THICK \I~\I ^ \\\ N za=4 1 N\ \ ^\ EM vs,VARIOUS Ru z 60 -- - -\w y \\ \\\ \^\"o ------ INTERPOLATED w (5wO 40' 0 I-I~I UJ 20 ---- --- -- ---- __ __ ----_N_^ ^ ^ ^ ^ ^ -- ---- _ 2 10 100 1000 10000 0, SCALE MODULUS FIGURE 21 REPRODUCED FROM REFERENCE 2

CafW=.-a 50 - NACA 65,-015 AIRFOIL.4-0. Z 0 \\ a, =40 w ' \L vs, VARIOUS Ru a..30 2 10 100 1000 10000 a. -, SCALE MODULUS IGURE 2 REPRODUCED FROM REERENCE 2 2 I0 I00 I000 I0000, SCALE MODULUS FIGURE 92 REPRODUCEFD FROM REFERENCE 2.

.7PI.6.5~~~~~~~~~~~~~~~~~~~o.4 0 3 OF~~~I ' —11111 II / W ~~~~~~~~~~~~~~~~~~~~le~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o 'o 100 ~~~~EFFICIENC Y OF RIME DEPOSITION ON oe ol oe.-loo, CYCLINYDERS WITH IDEAL FLUID FLOW 2 _____- ^ -% ' z / z..1~~~~~~~~~~~~~~~~~~0.1.2.4.6.8 1-0 2 4 6 8 10 20 30 40 60 80100 K FIGURE 23 REPRODUCED FROM REFERENCE I

.9.8 - E 7~111./~7~/~/7^~// / ------ 4001 0 60 80100 20 000 6 00 00 I \~~~~~~~~~~~~~~~~~~~~~~~~~/00 10.0,I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00 *s 7 '001 1 Ii/// ///// / / / / /r EFFICIENCY OF RIME 0EPOSIToON ON _ /// // // / T / ~7 / ^ ~~CYLINOERS WITH IDEAL FLUID FLOW K Of 01FIGURE 24 REPRODUCED FROM REFERENCE I. FIGURE 2EFFICIENCY OF RIME DEPOSITION ON

REFERENCES 1. I. Langmuir and K. B. Blodgett, "A Mathematical Investigation of Water Droplet Trajectories", General Electric Company Report, 1945 (also A.A.F. Tech. Report 5418). 2. A. G. Guibert, E. Jansen, and Wo M. Robbins, "Determination of the Rate, the Area, and the Distribution of Impingement of Water Drops on Various Airfoils from Trajectories obtained on the Differential Analyzer", University of California, Department of Engineering, Septo 1948o A. G. Guibert Addendum I to above, April, 1949o 28

SYMBOLS a radius of drop (ft.) CD drag coefficient" EM. total efficiency or total percentage catch; ratio of y at x =.O to radius of cylinder or sphere; ratio of distance between the initial positions of the upper and lower tangent trajectories to maximum thickness of airfoil* K defined by (4)* K defined by (5)* K0 defined by (8)* L Characteristic length (fto); radius of cylinder or sphere; chord length of airfoil RvU Reynolds Number based on velocity ato-,(2aUpa/ )* R Reynolds Number based on relative velocity of drop in air (RuI-vI/u)* 1S3 S furthest position of drop impingement on surface of airfoil, measured from chord line in units of chord length, on upper and lower surface respectively t time in units of U/LZ* U velocity of air at infinity (ft./sec.) U"xuy air velocity components in x and y directions respectively, in units of U^ u air velocity (ux + u2)1/2* v,v drop velocity components in x and y directions respectively, in X Y units of U v drop velocity (vx + Vy)l/ V drop impact velocity at stagnation point of cylinder in units of 1 x,y coordinates in units of L* angle of attack of airfoil (degrees) ks Stokes' Law range of drop in still air (ft.) * dimensionless 29

x\ ~ range of drop in still air (ft.) A't ~ absolute viscosity of air (lbs/ft-sec) 2 Ru/K p density of air (lbs/ft3) a Pd density of drop (lbs/ft3) ~9 angle measured from x axis of sphere or cylinder (degrees) IG' angle measured from x axis of cylinder or sphere beyond which no deposition occurs (degrees) V VR * dimensionless 30