Impingement of water droplets on wedges and double-wedge airfoils at supersonic speeds

28 Pages·2011·19.94 MB·English

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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
REPORT 1159
IMPINGEMENT OF WATER DROPLETS ON WEDGES
AND DOUBLE-WEDGE AIRFOILS AT
SUPERSONIC SPEEDS
By JOHN S. SERAFINI
1954
For ale b7 tbe SlIperinleDdent ot Documents. U. S. Go'fernmenl Printlni Office. Wa8hlniton 25. D. C. Year17 allbacrfplloD. '10; foI'e/iD Ul.2S;
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REPORT 1159
IMPINGEMENT OF WATER DROPLETS ON WEDGES
AND DOUBLE-WEDGE AIRFOILS AT
SUPERSONIC SPEEDS
By JOR S. ERAFINI
Lewi Flight Propulsion Laboratory
Cleveland, Ohio
National Advisory Conlmittee for Aeronautics
Headquarters, 1512 H Street NTl'., Washington 2.5, D. O.
Created by act of Congress approved March 3, 1915, for Lhe supervision and direction of the scientific study
of the problems of flight (U. S. Code, tiLle 50, ec. 151). Its membership was increased from 12 to 15 by act
approved Nfarch 2, 1929, and to 17 by act approved May 25, 1948. The members are appointed by the President,
and serve a such \\'iLhouL compensaLion.
JEROME C. HUNSAKER, Sc. D., Massachusetts Institute of Technology, Chairman
DETLEV \'l. BIWNK, PII. D., President, Rockefeller Institute for i\ledical Research, Vice Chairman
JOSEPII P. ADAMS, LL. D., member Ci\'il Aerollautics Boald. RALPH A. OFSTlE, Vice Admiral, LTnited States Navy, Deput.v
AI,LI"N V. AS'l'IN PII. D. Director National Bureau of Slandards. Chief of Naval Operations (Air).
PRE,;'!'ON R. BASSE'I"!', i\f. A., Prcsident, Sperry Gyroscope Co., DONALD L. PU'l',]" Lieutenant General, niLed SLates Air Force,
Inc. Deputy Chief of Starr (De\'elopmen t).
LI~ONAIW CAR~IICHAEL, PII. D., 'ecreLar.v, SmiLh,;onian Insti- DONALD A. QUARLES, D. Eng., Assistant Secretary of Defen e
Lution. (Research and Developmenl).
RAIJPIl S. DAMON, D. Eng., President, Trans World Airlines, Inc. ARTHUR E. RAYMOND, C. D., Vice Pre ident-Engineering,
J,uIES H. DOOLl'l''fLE, Sc. D., Vice President, Shell Oil Co. Douglas Aircraft Co., Inc.
LIJOYD HARRISON, Rear Admiral, UniLed States Na\'y, DepuLy FRANCI::; \\'. REICHEWERFER, Sc. D., Chief, nited States
and AssisLan t Chief of the Bureau of Aeronautics. \Veather Bureau.
RONA 1,0 M. HAZEN, B. S., Director of Engineering, Allison OSWALD RYAN, LL. D., member, Civil Aeronautics Board.
Division, General Motors Corp. NaLhan F. TWINING, General, United tates Air Force, Cbief of
Staff.
HUGH L. DRYDEN, PH. D., Director JOHN F. VICTORY, LL. D., Executive Secretary
JOHN W. CROWLEY, JR., B. S., Associate Di"ector for Research EDWA RD H. CHAMBERLIN, Executive O.fficer
HENRY J. E. RmD, D. Eng., Director, Langley Acronautical Laboratory, Langley Field, Va.
S~lI,(,H J. DEFRANCE, D. Eng., Director, Ames AeronauLical Laboratory, Moffett Field, Calif.
EDWARD R. SHARP, Sc. D., Director, Lewis Flight Propulsion Laboratory, Cleveland Airport, Cleveland, Ohio
LANGLEY AERONAUTICAL LABORATORY AMES AEHONAU'l'ICAL LABORA'rORY LEWIS FUG"'!' PROPULSION LABORATORY
Langley Field, Va. Moffett Field, Calif. Clcveland Airport, Cleveland, Ohio
Conduct, unde1' unified contl'ol, fo1' all agencies, of scientific 1'eseaTch on the fundamental problem of flight
II
ERRATA
NACA REPORT 1159
IMPINGEMENT OF WATER DROPLETS ON WEOOES
AND DOUBLE-WEDGE AIRFOILS AT
SUPERSONIC SPEEDS
By John S. Serafini
1954
Page 2, column 1: Line 4 should read: to 2.0; tangent of the semiapex
angle, 0.02 to 0.14; and
Page 12,- column 1, line 2: ~u and ~L should be I~ul and I~LI'
Page 16, column 1, line 5: In the equation Vi should be Vl'
Isd·
Page 19, column 2: The symbols Su' SL should be lSul'
Page 22; figure 11: The sublegends should read:
.
(a) Correlation of .
. . .
(b) Effect of .
NACA-Langley - 1-21-55 -1450
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REPORT 1159
IMPINGEMENT OF WATER DROPLETS ON WEDGES AND DOUBLE-WEDGE AIRFOILS
AT SUPERSONIC SPEEDS
1
By JOHN S. ERAFl [
SUMMARY impingem nL variable that mu t be determined arc Lh total
An analytical solution is pre ·ented/oT lhp equation.~ oj motion water catch, the extenL of impingement, and Lhe rate of im
qf water droJJ/et· impinging on a wedge in a two-dimensional pingemenL pel' uniL area of body mface. These variables
supersonic flow field with an attached shock wave. The closed can be deLermined analytically fl'om calculations of Lhe cloucl
jorm solution yields analytical expressions jor the equation oj droplet trajectories oblained foJ' Lhe variou aerodynamic:
bodie . Investigators have rcported the re uHs of studie of
the droplet trajectory, the local rate of impingement and the im
pingement velocity at any JJoint on the wedge sUiface, and the clouddroplC't Lrajeetories about l'igh -circular' cylinder (refs.
total rate qf impingement. The analytical expre ·sion are 2 to 5) and about airfoil (ref. 6 to 9) immersed in an in
utilized in the determination qf the impingement oj water drop compressible fluid. An evaluation of Lhe effect of compres
lets on the jorward ·Ulface· 0/ 8ymmetrical double-wedge ai/foil· sibility on Lhe droplet LrajecLorie abouL cylinders and airfoils
in supersonic flow field with attached shock wave8. up to Lhe eriLical flight Mach number is pt·e ented in refer
For a wedge, the re ·uLts provide injormation on the effects oj ence 10.
the droplet sizp, the free- iream illach number, the semiaJJex At pre enL, litLIe information exi t on Lhe impingement
angle, and the pre' ure altitude. For the double-wedge airjoil, of dl'oplets on aerodynamic bodies in a super onic airstream.
additional calculation proLYide information on the p.ffect oj air The con enLraLion of pa t effort on problem of impingemenL
foil thicknes . ratio, chord I nglh, and angle oj attack. on airfoil at sub onie flighL peed and the pre ent lack of
The Telsu/ts jor the symmetrical double-wedge ai/jail are also convenienL and rapid mean for obtaining Lhe rotaLional
correlated in terms of the total collection (fficiency as a Junction flow fields about airfoil at upersonic peed are pos ible
oj a relative modijied inertia paramfter. The results are pre explanations for the scarcity of tl'ajecLory calculations for
sented jor the jollO'l.u'ing ranges oj variable : jree-stream static Lhe upersonie region. An initial contribution to Lhe olu
temperature, 420° to J/JO° R; droplet diameter, 2 to 100 microns; Lion of the over-all problem of impingement of water droplel
free-stream 1I1ach number, 1.1 to 2.0; semiapex angle jor the on aerod}Tnamic bodies at uper onic speeds is given in
wedge, 1.14° to 7.9JO, and corresponding double-wedge-ailjoil l"Cference 11, which presents an analy i of the water
thiclcne ·-to-chord ratio, 0.02 to 0.14; pressure altitude, sea interception characLeristic of a wedge in a uper onicflowfiell.
level to 30,000 Jeet; and chord length, 1 to 20 Jeel. The present report extend Lhe analy i of reference 11
and further presenl an extensive tudy of lhe impingement
J TROD CTIO of water droplet on two-dimensional wedges and dOllble
The pr bl m of icc prevenLion on aircraft flying aL sub- wedge airfoil for super onie flight speeds that re ult in
onie speeds up lo crilicallIach numbel' has been a subject attached shock wave and constanL velocity fields behind the
of considerable Ludy and re earch by the JACA. The roe-enL hock waves. For the wedge ano-l and clouble-wcdgc-
adven t of aircraft flying at tran on ic and superson ic peeds airfoil Lhickne raLios considered herein, Lhe ho ck-w avc
ha required an exten ion of Lhe e icing tudies. That an aUachmenL ).Iach number varie from a value slighLly
jeing problem exist in the transonic and supersonic peed greater than 1 to abou L 1.4. The meLhod employed i ba ed
ralwe i verified in reference 1, which shows by an analytical on an analyLical oluLion of the eq uation of moLion by mean
invesLigation wiLh experimental confirmation that diamond of a closed-form integration. The closed-form solution
or ymmetrical double-wedge airfoils arc ubject to po ible yield analytical e}"'Pre ion for the equaLion of the lrajec
icing at flight Mach number as hio-h a 1.4. A imilar 1"e tories, the local impingement efficiencie , the velociLy aL any
suJ t is expect d for other airfoil shape being considered for point on the Ll'ajecLorie , and the total rate of impingement.
usc at transonic and uper onic flighl peed . Thi olution i made pos ible by the u e of an empirical
In conducting rcsearch on the problem of ice prevenLion relaLion for LIlt' drag coe[fteient for phere that gives a good
on aircraft and missile , regardle s of the magnitude of the approximation of lhe e:q)erimental drag coefficient .
llight peed, it j essenLial LhaL Lhe impingement of cloud The re ult of cal ulaLion for the rate, Lhe extent, and Lhe
droplet on airfoil and other aerodynamic bodie b deLer di Lribution of impingemenL of waleI' droplet on wedo-es
mined either by theoretical calculaLions or experiment. The and symmetrical double-wedge airfoil are presentecl herein.
I Supersedes NA A 'I' N 2971, "Impingement of Water Droplets on Wedges and Diamond Airfoils at Supersonic peeds," by John S. erafini, 1953.
1
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2 HEPOR'!' 115 NATIONAL ADYI 'ORY COMMI1vl'EE FOH AEHONAUTICS
The ntnges of variab!C'sinduC\('(l for the wedge arc: fl'('(' air-velocity fIeld ', whiGh in Lum prociuce idcnLiGal forcc
stl'cam statiG temperaturc, 420°, 440°, and 4GOo R ; droplC'l systems dowllstream of thc shock wave, ine pectivc o[ the
diameter, 2 to 100 microns; frec-stream \ Iaeh number, l.1 point along tbe shock wavc where tbe droplets cra s Lhe
to 2.0; tang('nL of the emiapex ano-le, 0.12 to 0.14; and ""a\'c. It follows, therci'ore, lllat, for droplets of a given
pre sur(' altitucie, sea level, 15,000, and :W,OOO fpel. 'I'll(' size, alllhc trnjedories ill a given problem nrc idcnLical with
ranges of variables [01' the douhle-wcdg(' airfoil are the same I"C pecL Lo t.he poinL where thc droplet crosse lhc hock wavc,
as Lhose [or the wedlre, alld the additional yariabh' for lhe By adopting a frame of r('/"e1'C11Ce lIntL mo\'cs aL the C011-
.nl1l1letrical double-wedge airfoil range from 0.02 Lo 0.14 for stant vcloeiL,\" of the ail' 1 '2 dO\l11strea111 of Lhe shock ,,"ave,
tbe thickness ralio and from 1 to 20 feel for the chordlcnglb. the problem of the droplet motion is rec1ueed to the still-air
The ""ork reported herein \I'as performed at lhe NACA problcm, defincd as thc dcLermination of Lhe moLion of a
Le,,"is labora lory in the spring and ummel' of 1952. dropleL lhaL, having an iniLial velocity, is projccLed inLo
quicsccnlnir. n cnee, rclali,'c to the moying frame o[ refer
A ALYSIS
STATEME 'T or PROBI.EM ence, Lhe initial "elocil,\" of Lhe ciroplet upon (Tossing thc
The solution o[ the problem o[ impingemenL of wale1' siwek ,,'a,'c is cqual to the vectorial ciiJl"ercncc of Lhe frce
'-2
dropleL 011 a t"'o-dimen ional wedge al super onic speed stream ail' vdocity VI and the ail' veIocity dml11 tream of
with an attached shoek wave is not as difficulL a thaL for Lhe sho(']';: \\'a vc. Adoption of tltc framc of reference moving
thc imping('mcnt on variou airfoil at low subsonic peecls. wiLh a con tanl velociLy ]' duces thc problem from thc olu
For the wedge aL supersonic pecds with an at tached shock Lion o[ h\"o imullaneou nonlincar sc 'ond-order differcnLial
waye, 1I1(' ail' yelocit)" eyerywhere ahead of the shock wave equations in Lhe fixed coordinate sy tcm to the olution of a
is constant and equal to the fre('-stream air velocity rJ single nonli ncal' econd-order clifrel'en Lia1 equation in the
(fig, 1). Thc air velocity behind Lhe shock wave 1r is also moving coordinate system. AI, any given insLant, tbe drop
2
e\-erywhere consLanL and parallel Lo the wedge urIace. All lel displacement relativc to Lhe poinL o[ intersection wiLh the
Lhe droplets llave the amc inilial velocity (lhat of the free shock '\'nve in the fixed frame of reference is obLainecl by
stream air yeIocity), and lheir LrajecLorie arc exactly co adding YCdorially Lhe droplct displacemcnl wiLhin Lhe mov
incident with thc air Ll'eamlines up tream of the hock wave. ing frame of referencc Lo thc eli plaeem nL of Lhe moving
All ,,"ater droplC'ls of a givcn size a]" subjecLed to identical framc of refcrcnce for Lhc ame i.ncremen L of time,
At 1=0 At I = I
-
~V. . iZ -u- -Tu -~--
(J" (f
_ __(J_" __K ..=Lf T_-<1_"'
VI
Droplet
Droplet
trajectory
trajectory
s
---------------------------------------------------------- Wedge
center
line
FIG CRE 1.- < chcmaLic diagram of \yaLer-dropleL LrajecLory for wedge in supersonic now \\"iLh attached shock wave.
3
IMPI TGEME T OF WATER DROPLETS ON WEDGES AND DOUBLE-WEDGE AIRFOILS AT SUPERSO Ie SPEEDS
This general meLhod was used in reference 11, 'where the Lhe droplet emerge wiLh a v 10ciLy VI' In Lhi case Lhe
one second-order differential equation representing th drop ini tial velocity of the droplet is
let motion relaLive Lo Lhe ail' velociL)Tb ehind the hock wave
was in tegrated graphically. However, it is po ible Lo
obtain a completely analyLical solution by means of a closed
which i lhe magnitude of the vector clifi'erenee of the air
form integration without re orting Lo tbe use of numerical
velocity vector upstream and clown tream of the aLtached
integrations or analog computing equipment,if an empirical
shock wave. As can be shown from consideration of the
relation i assumed for the drag coefficient as a filllchon of the
contirmity equaLion and Lhe equation for can ervation of
Reynolds number of tbe droplet relative to the air'. It will
momentum acro s the oblique hock wave, the velocity ve -
be shown that this clo ed-form integration of the stiU-air
tor U i normal to the hock wave. At uny ubsequent
problem when applied to the wedge in super onic £low with j
attached shock wave yields the equations for the trajectories instant of Lime, the relative droplet velocity vector U l' tains
of the water dropleL and the droplet velociLies at any point the ame' angular orientation to the shock wave and changes
on the trajectories and make available relation for the only in magnitude.
rates of total water impingement and the local rates of water In reference 11 lhe solution of equalion (3) is obtained by
impingement along the 'wedge uJ'faee. Furthermore, it is numerical integration. The)'e ult obtained in thi manner
shown that the e equations can also be readily applied to the makes it necessary to u e a graphical proeedllre in deLermin
determination of the droplet impingement on a double-wedge ing the trajeclories and the local rates of impingemel1 t.
air'foil in supersonic flow with attached shock wave. However, an analytical olution of equation (2), which elimi
Three of tho u ual a sumptions made in the pl'eviou in nates the graphical procedllre, can be obtained if the experi
vestigations on impingemenL at subsonic peeels and also mental vallles of the drag coefficient OD are e:\.'})re sed in a
required for this invesLigation are (l) Lhe wa tel' dropleLs arc funclion involving th \ Reynolds number ReT' The )'elation
always spherical and do not change in ize, (2) Lhe force of lR
gravity on the droplet may be neglected in comparison ,,-itlt
(5)
the drag forces, and (3) the drag of the ail' on the droplet is
that of a vi eous incompressible fluid. Here it is additionally
where ~ and marc lhe empir'ical consLants. This empirical
a sumed that (4) the two-dimensional supersonic flow field
relation i a valid approximation in the range of Reynold
about the lI'edge is fl'ictionles except within the infinitesi
numbers to which cloud droplet are ubjectecl in trajectory
mally thin attached shock lVave, (5) no condensa tiOD shock
calculations. llbstitulion of the expres ion for OD (eq. (5))
occurs and ne change in pha e occurs as the water droplet I
in equation (3) resulL in Lhe e:-"'})l'e. ion
tmvel'se the oblique shock wave, and (6) Lhe unbalance of
Lhe forces on the water clroplet from the in ta,nt it enters the
dU =d2:J:=_~~ U [1+~ (2P2Ua)mJ (6)
shock wave until it emerges from the shock wave can be Deg dt dt2 2Pwa2 J.l.2
leeLed in the calculation of the Lrajectorie .
where the local relative Reynolds number Re = 2pzUa/J.l.2.
r
EQUATION OF DROPLET MOTION I MOVING REFERE CE FRAME The di placement of Lhe waLer dropleL in the moving frame
The velocity of the dropleL in the moving frame of Idel' of reference x is measul'cd hom the air streamline Lhat inter
c.nce is secLs the shock wave at the point where the water droplet
enters the air-flow field downstream of lho bock wave.
(1)
The closed-form integration of the difi'el'ential equation (6)
Va i presented in appendix B. The usc of 2/3 for the e}"'Ponent
where is the droplet velocity with respect, to the fixed
m and 0.158 for the valuc of tbe empirical can tant ~ in equa
frame of reference, and \1 is the ail' velocity downstream of
2 Lion (5) yields an ompirical curve fol' tbe drag coefficienL as
Lhe attached shock wave also wiLh re pect to the fixed
a function of Lhc local Reynold numbers Lhat approximates
frame of reference (fig. 1). In the frame of reference moving
vcry well the variation of the experimental value' of the
with the velocity \1~, the statement of Newton's la,,- of mo drag coefficient in the range of Reynold number from 0.5
tion for the water droplet become to 500. The value of 2/3 for th ex.--ponent m also facilitates
the closed-form integraLion of Lhe di.frel'enLial equation of
D = - 0 D 21' pz7ra zU'- =34" tra 3 pw ddUt (2) moLion. In figure 2 a graph of Lhe empirical relaLion is
pre ented, along with the drag-coefficient data of referencc
4 and 12.
from which
The l'esulls of Lhe inLegl'aLion are given by the following
2
dU =_~!!.'!.. U OD (3) equation:
dt Pw a
x=!!-. ~-3/Z Pw [Re )/3~)/2+Lan-) (Re -)/3~-I/Z)_
(A complete list of symbols is given in appendix A.) 3 P2 r, j T, j
Equation (3) i the differential equation of motion of a
droplet projected with an iniLial velocity into a region of
quiescent air' (the so-calle 1 till-ail' problem). The hock
wave is considered to be a sUTface of discontinuity from which
4 REPORT J 159- NATIONAL ADVISORY COMMITTEE FOR AERO A TICS
RELATIONS REQ IRED FOR APPLICATION OF CLOSED,FORM SOLUTION
U= ReU ie 3/2 [(Rer ,; -2/3e-l+ l) eT_ l)-3/2 ( ) TO OBTAIN DnOJ>LET MOTION AND IMPINGEMENT IN FIXED
T, i REFERENCE FRAME
and Impingement on wedges.- For a jJroblrll1 of given aero
dynamic eonclitions, thr trujectorirs of all thr I\'fitrl' droplet
(9)
of a given si;-;e arr iclrntieal wlIen the points w!lrre Lhe droplet
t rajectorie intrrseet tlIe hoek wave firo superimposed.
This uniqur characirl'istic of thr watrr-c1l'Oplrt trajecLorir
O <cp<~ about a wrcigr in uprrsonic flow with an ailfichrcl oblique
= = 2
hock "'avr is the re ult of two constant yrlocity field , one
upstrram and on(' downstream of the shoek \\'a\'r. There
Thr intr1'11lC'clifttr steps of intrgration are gin n in appendix E.
fore, only one set of equations for a single trajeclor)' is neces
Equations (7) and ( ) givr, re 'pectivcly, the displacement
ary to caleulatr [.hr impingement parameter for a specified
anclthr velociLy of the droplrL at an~T instant in the moving problem, including a given droplet size. The yalue o[ the
framr of rdC'rence. The displacemenL of the droplet with
initial rclfitivr veloeit:I' Ui, Lbe initial Reynold number
rrsprct to the point where it crossr Lhe hock wave can be
ReT,;, and Lhe ([ellsity ratio Pw/P2 arc Ilrecled for ubstitution
obtainrd hy ft vectoriftl ftclcliLion of the displacrment x and
in the closed-form olution of the equatiolls of dropleL motion.
the clisplacrment of the moving reference frame in Lhe coJ'
The e value can br obtainrd from information available in
Va
]'rsponcling Lime interval. Thr droplet velocity relative
reference 13 ancl from thr usc of impie algebraic and tl'igo
to thr fixed frame of rcfrrence must al 0 be obtained by the nomrtric relaLion for given yalues of lhe free-slream tatic
YC'ctorial addition of V (eq. ( )) and V2. Equation (9) temperalure tl, the droplrt diameter ci, tilr h er-sLr am ;"Iaeh
giYC's thr maximum "alur of ,I: obtained a the time of tra\"rl number J[I, the angle of surfaee inclination to the free-stream
In tlIr air-nolI' field down treftm of the slwck wa\'e direction IT, and thr fr('r-strram slatie pn' lire PI. These
appronclles infinity. The significftnce of this quantit~, will rrlations resulL in the [oUo\\'ing exprr siolls for the inilial
br di cllssed in subsrqlJent ection. rclative ycloeity anll thr initial relatin Re~'nold !lumber:
20
vy
v
10
./
V
I /'
6 I y-
Data
o Williams (ref. 12) ~
0 Langmuir (ref. 4) I/V
3
/v
V
I
- 2
, /
V
a':":' -I <;t
/'
~(\J I
- - - t-I" /
-
c
'" ./
:~ .6
., ./'
o ./ I
U
C0oi ' .3 V / I II
!
/
. I
.2
V
. ,
/ V
./I
.1
./
L
.06
/
./
V
10 100 1000
Droplet Reynolds number, Rer
FICI'RB 2.-Compari~on of cmpirical rclaLion for drag coefficicnt as funcLion of clroplcL H.<'ynolcb numbcr \\"iLh cxpcrilllcnLal valucR.
5
IMPI GEMENT OF WATER DROPLETS ON WEDGES AND DO BLE-WEDGE AIRFOIL AT SUPERSONIC SPEED
(l0)
(15)
and
ince x is a function of in equation (13) and (15), tbe
(1J) T
expression for.l and ,J'(, pec[ivdy, ar fUJlc{'ions of T.
However, inee ('annoL be eliminated from equaLion (13)
T
A convenient form of the solution for the impingement on and (15), S cannot be obtained expliciLly as a funcLion of r.
a wedge 01' the front half of a double-wedge airfoil is obtained 1 everthele , the curves of .I, Lhe initial displacem nt of the
if i defined a the distance to the point of impingement
water-droplet trajector.lr from the leading clge normal to
mea w'ed from the leading edge for a water droplet LhaL
ihe free-stream direction, again t S, Lhe di tance to Lhe point
enter the £low field behind the hock wave at a distance r
of impingement of the tated waLer-droplet trajecLory, can
above the leading edge (fig. 1). The unique relation between
be obtained by ub tiLution of the ame et of value for
T
and r in a given problem for droplets of the same size i
in Lbe CA-pre ions for rand S.
quite readily determined by considering the displacement of
An analytical expre ion for Lhe local impingement
Lhe waLeI' droplets a Lhe vectorial wn of Lhe displacement
efficiency can b obtained from the preceding cxpre ion
of Lhe water droplet relative Lo Lbe moving frame of reference
for rand S. The local impingement efficieney {3 is defined
and Lhe di placement of the moving ference frame reI a Live
l' by the eA.'})r s ion
to a fixed frame of reference (referred to wedge). • ince the
moving reference frame ha a velocity equal to tbe air velocity
(16)
V which icon tant in magnitude and parallel Lo the wedO'e
2,
surface, only Lhe droplet travel in the moving reference frame
include Lhe component of dropleL Lravel representing the
approach of the waLeI' dropleL 1,0 Lhe wedge urface. For a wh 1'e ~r i the difference in the initial eli placement of two
waLer droplet tarting from poinL A and impinging on the waLerdroplet having VNy nearly eq ltal initial displacement ,
wedge urface aL point 0 (fig. 1), Lhe di placement of the and ~S i the mall increment of wedO'e urface beLween Lhe
moving reference frame (= V t, wbere t is zero at point A) j point of impingement of Lhe two water droplet. Differen
2
given by the di placement veeLor A B equal to CD, and Lhe LiatilJg rand S wiLh 1'e p ct 1,0 T and performinO' Lhe division
droplet motion in the moving frame of reference is given by indicated by equation (16) yield the following expre iOll
tbe di placemenL vector BO cqual 1,0 AC. Therefore, rela for (3:
tive to the starting point at A (fig. 1), the di placcmen I, of (17)
the watcr droplet to Lhe point of impingement at 0 i
obviously equal to AB+ BO or to AC+ CO. From figure 1,
where
the eli placement of the water droplet aL the point of impinO'e n = in e tan (lI+ cr) (J a)
ment 0, mea ured from Lhe leading edge at E, is given by l
adding Lhe vector EA 1,0 the displacement vector from the 112= ('c (lI+ cr) (l b)
Lat'ting poinL A; and the displacement of the dropleL at 0
referrod Lo Lhe leading edge is (l c)
s = IEOI= IEA+ AC+ COI= IE C+ COI= IECI+ ICOI=~+{ Since {3 and both are function of T, the local impingement
( 12) efficiency {3 at any point on the wedge surface i delermined
by using the same value of in equaLion (17) and (15). The
T
where ~ i the magniLude of the displacement vector EC, value of {3 that ('xi t a the point of impingemenL of the
and { (= V t) is the magniLude of Lhe displacement vector \\-ater droplet on the wedge uJ'fa,ce approaches the l('acling
2
CO (the di placement of the moving frame of reference). edge as a limit (S~O) i given by the following:
The values of r and ~ are obtained in Lerm of x, the di tance
of trav 1 in the moving reference frame, from simple trigo III cr (l D)
nometric identities involving the various angles shown in
figure 1:
r=x sin e tan (lI+ cr)
The magnitude and direction of the droplet v('lociLy aL tb('
wbere x i given by equation (7); by point of impinO'ement Vd• tm (relative to the fixed frame of
II reference) can al 0 he ea ily obLained aL any point on Lhe
w dge urface a
(13a)
and V d. im = 1'7 I -y/ "0 2+ W 2( UUlm)2+20__ W (UJtm co (II+v~ ) (20)
~=x ('e (lI+ cr) (14) t t
(+ )]
[Uim .
ub tituLion of equaLion (14) inLo equaLion (12) for the ur Ki",= cr-cri, ,.= cr- S.l n - I V Sin II cr (21)
d.tm
face distance to the point of impingemen t yield the following
equation: where i the angle between the free-stream velocity vector
K
REPORT 1159- ATIO AL ADVISORY COMMITTEE FOR AERONAUTICS
VI and the droplet-velocity vector Va, and is Lbe angle
(I'
beLwcen the droplet-velociLy vector Va and the air-velociLy
vector V In equaLions (20) and (21) for a given Lrajectory,
2•
VI, £2, w, U;, II, and (I are con tants. Therefore, for a given
trajectory, Va. and arc funcLions only of U (eq. ( )),
1m Kim
which is in turn a function of the dimensionless time
T,
-- variable.
AL-------------- Impingement on symmetrical double-wedge airfoils.
The impingement on a double-wedge airfoil may be obtained
~
-- -----!-1----- from the solution Lo the problem of impingement on a wedge
--{-_-.:_- 0
as presented heretofore. In this report, the double-wedge
airfoil considered is ymmetrical, the maximum thickness
occurring at 50 percent of chord (fig. 3). At zero angle of
attack, the impingement on a double-wedge airfoil will b
limited to the region from the leading edge to the shoulder
at 50 percent of chord. The solution for impingement on a
wedge sm-face having a given semiapex angle (the angle of
(I
inclination of either wedge surface to the free- tream direc
Shock wove Lion) can also be used as the solution for a double-wedge air
( 0)
[oil where the thickne s ratio is equal to Lan the tangent of
(I,
;."E'~"p onsion zone the semiapex angle, and ,,-here the ch·oplet size and other
/ 'y/ // parameter o[ the problem arc the same flS for the wedge.
Therefore, the values of the local impingement efFiciencie
/ /
/ / f3 and f30 at any given point on the surface will be identical
/ /
I / for both the wedge and the double-\\-edge airfoil at zero angle
I /
of atlack under lbe aforementioned similarity of conditions.
1//
A --------B -1-/ The solLi tion for the impingement on the double-wedge
~ airfoil at angle of attack can also be obtained from the
solution for impingement on a ,,·edge as for the case of the
-~ double-wedge airfoil at zero angle of attack. l'Vhen the
a
C~r ---.------~~O symmetrical dOLI ble-wedge airfoil is at angle of attack a,
\\ the angle of inclination of its forward upper surface to the
\\
\\ free-stream direction is eq ual to (1-a and that of the forward
lower urface is equal to (I+ a. Therefore, the solution to
\ \
\ \ the impingement on the upper and lower sLirfaces of the
\ \ double-wedge airfoil i obtained from the solutions for
\ \
~\ impingemen ton wedgos having the redefi.ned semiapex angle
o[ (I- a and (I+ a, respectively, where the droplet size and
Shock wove E~ponsion zone
other parameters of the problem are kept the same. For
(b)
the double-wedge airfoils at angles of attack having tangents
equal to or greater than the thickness ratio, the water
droplets will not impinge on the upper surface. At angle
of attack baving tangents greater than the thicknes ratio,
some water droplets may impinge on the lower mface
beyond 50 percent of chord. The e three conditions are il
lustratecl schematically in figure 3. For a< tan-I(Tlc), the
----
impingement occurs on surfaces AC and A B; for a= tan - (Tic),
---- I
impingement occurs only on mface AC; for a> tan- I(Tlc),
impingement occurs on lower urface AC and may occur
c
\\ 0 on lowol" surface CD. However, tbe condition where
\ \ a> tan- I(Tlc) i not considered herein, ince the solution
\ \ pre entee! 111 this report is not valid for the determination
\ \
I \
(a) Angle of attack cx< tan-J(Tjc).
\ \
\ \ (b) Angle of attack cx= tan-J(T/c).
\ 0 (c) Angle of attack cx> tan-I(T/c).
Shock wove Expansion zone FIGURE 3.- Schematic diagram of ,ymmetrical double-wedge airfoil at
(c) angle of attack in supersonic flow wiLh attached shock wave.