It is obvious that, in accordance with Newton's second law, the lift on an aerofoil must be equal to the vertical momentum communicated per second to the air mass affected. Consequently a lifting aerofoil in flight is trailed by a wash which has a definite inclination corresponding to the factors producing the lift. It is thought that sufficient data, theoretical and experimental, are now available for a complete determination of this wash with respect to the variation of its angle of inclination to the originating aerofoil and with respect to the law which governs its decay in space

Diehl, Walter S

English

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TE.C~ECAL lTOTES 
~A:':;:C'JA~ LDV:;:SGRY G~i1IITTEE FOR AEOONAU':'ICS. 
No. 
L't. :: allJer S. Diehl ~ 
13ur e at: of Con st ru.c"';; i Oj1 & Rep.'1i:', J". S. K. 
----~, -----...-
J?.l:uary, 1931. 
https://ntrs.nasa.gov/search.jsp?R=19930080891 2020-03-17T04:17:21+00:00Z
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NATIO~JAL ADVI:'-C'_t"i CO~nj=TTE:E FOR AEROI AUTICS. 
THE DETERM :!:NAT ION Or.' DOVlNWASH. 
B--j 
Lt . Walter S. Diehl, 
Bu_::eau of Const ruct ion & Rei.)air) :.., . ;: 4:'T • 
11-1;:r odl.~ct i 0:::1. 
It is ob--:l,ous that ) in a cco"C"'aance .1ith lJe~1to!11 e sec-
ond. la-'7, "Ghe lift on an aerofoil must be equal to the ver-
t iC?al mor:~er:tur.l c om:.1Uc-:. ic[.ted. ?er second to t~le 3.ir TQc..os 
affected . 0021sequentl: a li::ting aerofo':"l in f_ight is 
trc..iled. by a ',7ash 7vhich :1as a ciefinite ::"nc:i::. auio:: COl'!'e2-
ponding to the factors produoing the lil~. 
that sufficient data, ~heoretical aLd experime~ual, a:re 
now available for a complete det e r::dna""Gior~ 0= thi8 ',;-as:'"l 
wit~ res~ect to the variation of its a~gle of in~lination 
to the originating aerofoil and wlt:: res)eet "GO 'uhe 13.',7 
which governo :ts dsea' in space. 
l:un:'.:<:1 s Fornula :or Do,mwas: . 
Al t~1ough it has 10:::lg been y-no'~ t ha-: the aLgle of d.o -.1'_-
wash () as obse:rveci 3.t a given poir:t behinj the aerC-
foil, is d~re0tly propcrtional ~o une ll:t of ~erofo~: 
(Br . A. O. A. R. c-:. 1L 1\'0 . 196 ) and inversely proportiom:.l to 
the aspect ratio (Lancllester IIAerial Flightt' Vol. 1, Chap-
- 2 -
• tel' VI II) Br . A. C. A. ?L. & M. No. lSl), Munk (Tecn!lisc;:'e Ber-
ichte III- l) - seems to have been the first to propose a quan-
titative solution . He assG:o:-ts tha-: € r.mst be repre sented 
as the product of some constant and. th0 angle of attaci\: as 
expressed "0)' the fortmla (Betz, T. B. 1-4) J 
57.3 
TT • 2 Lc s (-- ) 2_ b a 
',vhere b is the span) Lc is the lift coeff:i.cient and S is 
the area of "Ghe aerofoi l. The formula for do-;:nrlash then 
becomes 
5-?3c 
=-- 2L ( ~,,) 
. c '" b 
(=c . a. Ti 
the -.rai.ue of the constant c being de-cermined :,y experi-
mer..t . The formula as given c:.pplies to ~". onoplane s "m1"c ~:'c...--
be appl ied to Eul t iplane s , according t o Munk, ".Jy tta int 1'0-
dl:ction of another constant k whi8Ll reduces "Gne span b 
to the span of the equivalent monoplane. 
The values of 0 were determined fa:>:' seve ral mocels 
by photographing a seYies of strea~ers. O:ring tc th~ lac~ 
of cer-cain vital cata, the Tesillts nave not bee!l inc~uded 
in this study) "out the conclusions are giv-en i~lstead. It 
appears that the equation , as given above, is nOli general. 
The values of c va~y some~hat ~ore ~ha~ is allo~able for 
a 11 constant II . No attempt WUs mad.e to determine the varic~tion 
of wi th aspeot rat iO, nor was any allowance mad.e for 
the inevitable dying out or tIle viash effect i::1 space. It 
appears , however, that the ang:e of down~~sh is suostanti~ 
- " -
ally constant over about eight-tenths of the span , with 
sudden changes near the tips . 
N. P . L. Formula. 
The rr.ost comprehensive series of tests on dOWTHio.sh, 
which have been publi sne d, are those by Sandison, Glauert, 
and Jones (Br . A.C.A. R. & M. No . 426) . In this ir.veGtigation 
the variation of dovmwash was determined in space for a 
nur.:ber of points behind, above, and below the trailing edge 
of the aerof oil. It 7vaS found that, in accordance with 
hydrodynamic theory, * tne angl e of do~-mv,ash decreas9s ex-
ponentially wi th the distance fro~ the aerofoil (a ~i-
~lane in this case) and mi ght be expressed by the vK;iri-
c 5.1 f o T1ilU 1 a 
where 
E: = (0 
10-0 • 0 5 ~ - 0 • 0 8 , 
is the distance behind the wing in chord lengths, 
is the distance belOW the chord of the upper wing 
iE terms of the gap, 
and (0 is a constant for any given arrange~ent. 
This appears to have been the first atterr.~t to express 
the variation of ~ from po int to poin~. With ~ sntis-
factory law for the v~riation of E:o it ~muld have oeen 
compl ete. 
Derivation of a Comprehensive Downwash Fo~mula. 
It is now possible to derive a comprehensive do\m~ash 
formula based on the G~ttingen t heoretical and the IT.P.L . 
* See Lanchest er , Aerial Flig~t, Vol. I, Chap. III. 
4 --
empirioal formulae . It is k-uOVlln definitely 'Lhat dc',iIll'"Ctsh 
var ies exponen t ial:!.y wit h di st ance from the trail ing edge . 
The plo-;: -ueci re sults of N . P . !" . inveatigations ) v7hich shOi:V 
tfiis variation vertically and horizontally, are ~iven in 
Figs . 1 and 2 , respectively . The data in Fig. 1 ~ave been 
repl otteci on a logari thr.Ll ic sC 2.1e in FitS. 3) '''lit:l -~:le ver-
tical distanoe from the trail:'..ng edge expresseci in c~lo::cd. 
lengths plus one chord. J. ength* as abscissa and angl'3s of 
dO lJVIl wash as ol'ainates . It is found that for a giV'en angle 
of attack, the angle s of cio -rm-.7as~ at 'J'ariollS vertical dis-
tances from ·Ghe t ~ail Llg edge l ie or.. a straight line. 
The li~es cor responding to the various angles of attack 
are a.ll parallel and have a slope of -:13c ± (). 5°. Thi::> 
indioa~es tha~ the variation of an6le of downwash \~ t~ ver-
tical dista~ce from the trailing edge oan be represen~ed 
by an equation of the form : 
(..=c y + l;1: , 
2 
·whe re 0 1 = a. oonst ant, 
y :::: the vertioal dist9.nce, of the :Joint under- oon-
side rat ion, in oh8r~ length3 , from the trailing 
edge J 
and n - tan (-130 ± 0 . 5°) 
= -0. 83 ± 0 . 01. 
In a similar aanne r the data from Fig . 2 have been 
plot t ed in Fig . 4 . The p oints again fall near parallel 
straight lines b ut their slope, - 21 0 ~ 0 . 5°, ic steeper 
than that in },'ig . :) . The indioated variatiOlJ. of the angle 
* This is necessary so as to ~rovide a finite value at the 
trail ing e dge . 
- 5 -
of e.ovmwash with variation of horizont::;.l dis"':ance from 
t~e tr~iling edge is of the form 
and 
€ = C 
:2 
x: = the hJ-.:'i:::on-:'al distan:::;e) cf the point under con-
s:_dera:':;io:l in c hord l enbths _, from the trailing 
e ge , 
= - 0 .33 ± O. '11. 
In order to eli~inat e t~e c~lcul~tions involving frac-
tional ex;c~ents ~he fULctions , 
and 
z = (y + 1)-0-23 
Z = (x + 1)-0-38 
have been evaluated and plotted in Fig . 5 . 
Do..ta fron: five series of dO·.-m·.-c.,sh ciet3rr::in3.tions have 
been 'plotted in Fig . 6 , with angles of do~m:-:ash as cr::ii-
nat es and lift coefficients as abscissae . 7he slope of 
the straight line, \.hich passes through the points repre -
senting a seTies of :tests , deter:Y.ines the value of 6(/6Lc 
for that particular arrangement ane. the point in space at 
which the observations Ivere taken . Tl1e aspect ratio, ~he 
value of 6(/6[c' and ~he coordinates of the ocse:~aticn 
point are given, for each series of tests, ir. Table I. 
It is evident from ins;ection of F:g . 6 that (. var-
ies directly with lift ·coefficient. It has also been ShCWi.l 
by data from the tests of Sarldison, Glauert, and. Jones 
(Br . A. C.A . n. & ~,1. IJo . 426) how E: varies i:l space . :~UY:kl s 
- 6 -
eq'l<!tion indicates that e: ~.ra.rles iilversely aa tr..9 2..SI=3ct 
ratio., n . Therefore -ehe angle of d01j;n\T:::.i.sh sLould be given 
by 
-Nhere K is a ccns-:;aLt> numeric2l1J- e2.lu3.~ to D.e:/6Lc at the 
trailing edge of a vying of aSl)6ct ratio unity. 
'rhe value of K is G.ete!'mined for each of the five ser--
ies of tests ,Vl1Tic :l at'v plotted in Fig. 6, "0y su'ostituting 
the pro}ier values for the functions of x and y and for 
the aspect ratio n. The procedure is indicated by tee 
headings of columns in Table I . 
It is found that K is substantia::"ly constant, varying 
from 164 to 176; a single exception of 158 corresponds to 
a series of tes-cs on a bip:ane arranger .. em~, the wings of 
wtic: 17ere equipped -.vith flaps and ..Le:r:-reaent abnormal con-
ditions. It therefore appears that the angle of down~Rsh 
can be represented to a good approxirlati:m by 
e: = 1:0 . (x + 1) - 0'38 
E: = 170 F F L n . X· y ' c 
Fx ~nd Fy being the values of the func~ions of x and y 
which are given in ~ig . 5. 
The validity of t l is fOTr..mla :'s obvicusly con:ined 
to that range of angle of at~ack or lif~ ccef:icient 'n 
which tile air flow about the aerofoil is not abno:'~[:1c.ll:r tur-
bulent . 
- 7 -
APPlication of the Do~mwash Formula. 
The chief use of a dOWDlmsh formula is the calculation 
of the aerodynamic angle of attack of the horizontal tail 
surfaces. For this · purpose a reference poi~t is taken on 
the leading edge of t he horizontal tail surfaces and the 
values of ~ obtained from the for~ula. The aerodynamic. 
angle of attack of the tail surfaces will then be 
%=CL-~-e: 
where a is the angle of attack of the wing and ~ is the 
acute angle between the cnord lines of the \nngs ani hori-
zontal tail surfaces, considered positive (in the equation) 
if the tail is set at a le ss apparent angle th3.n the wings . 
The data from tests seem to indicate that iL case 
of a biplane the :aximum angle of dov1:lwash occurs in "the 
horiz ontal plane midvvay bet ween the t iVO wings. T~le e ffe ct 
is so slight, ho·;vever) that the above Lletho0.. may be used, 
referring the coordinates of the refeyence po int to the nea~ ­
est wing (preferaoly to t e no lift line), ':~ith the assur-
ance that the re3ults so ob"t&invd nIl be as precise as 
it is pract icable to calculate them with the data nO~1 avail -
able. 
- 8 -
TABLE 1. 
DETEm.nNATION OF K IN THE EQUATION 
K ( = -
n 
Source of:Asp~ct: 
Data natio: ~~ x y 
6Lc 
n 
N C-l . 
" 
Curtiss 9 . 5 :11.20:2 . 5 : 0 :. 625:1.00: 17.9 170 
Tests 
R&M #196 6 :1 5.20 : 2 . 3:0 . 55: . 64 . ,90: 26.4 158 
R&M #426 6 :14 . 70:3.0 :0 . 61 : . 595 : .89: 27.8 167 
R&:;1 #426 6 :16 . 00:3 . 0 :0 . 43: . 595:.915: 29.3 1'76 
R&£~ #515 7.73 : 11 . 8 :2 . 6:0 . 6 : . 62 : . 895: 21. 3 164 
NOTE : There wil l be a slight decrease in the valu3 cf 6( /6 Lc 
with increase in lift coefficient if t~e reference 
point is not fixed i n space. This is caused by "the 
c hange in the coordina~es of the l)oint with change 
in angle and t~e effec t may e~sily be accounted for. 
then 
Let D = cii stance from -'crail ing edge, 'l', to re f-
er2:mce point P 
e = Inclination to t.orizontal of line T? 
x = D cos 8 
y = D sin e 
n 




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The Determination of Downwash

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