A formula for the computation of the vertical velocity component on all sides of an airplane is deduced and discussed. The formation is of value for the interpretation of such free flight tests where two airplanes fly alongside each other to facilitate observation

Munk, Max M

English

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TECHNI CAL NOTES 
nATIO~AL ADVISORY CO_.tHTTEE FOR AERONAUTICS. 
NO n 216 
T~1E VELOCITY DISTRIBUTION CAUSED 3Y AN AIRPLANE 
AT THE POINTS OF A VERT ICAL PLA NE CONTAINING THE SPAN. 
By Max H. Munk . 
~arch, 1925 . 
https://ntrs.nasa.gov/search.jsp?R=19930081015 2020-03-17T04:24:56+00:00Z
NATIONAL ADV:tSORY CO~irliITTEE FOR AERONAUTICS. 
TEcaNICAt NOTE NO. 216. 
THE V~LOCITY DISTRIBUTIGN CAUSED BY AN AIRPLANE 
AT THE POINTS OF A VERTICAL PLANE CONTAINING 
THE SPAN. 
By Max M. Munk. 
Summary 
A formula for the computation of the vertical velocity compo-
nent on all sides of an airplane is deduced and discussed. The 
formation is of value for the interpretation of such free flight 
tests where two airplanes fly alongside each other to facilitate 
ob serva tion. 
,In certain kinds of free flight experiments with airplanes, 
a second airplane is used for the observation of the airplane to 
be tested. For instance, during some experi ments of the flight 
section of the National Advisory Committee for Aeronautics, two 
airplan~s were flown side by side. A cylindrical weather vane was 
suspended under one of the airplanes. An observer in the second 
airplane took photographic pictures under different conditions of 
flight in order to determine the angle of attack. The distance 
of the two airplanes was sometimes as low as one wing span. 
n .ll.a.A. Technica1 Note IJO . 216 2 
For the interpretation of such tests it is desirable to be 
informed on the vertical ve10ci ty component of the air caused by 
the observine; ai r plane in the region occupied by the airplane to 
be observed. Thi s information i s contained implicitly in Refer-
ence 1. For Greate r convenience I procecd to lay down in this 
Note the direct way to obtain this downwash. 
Tho problem is, then , to determine the downwash sidc~isc of 
of an airplane in a moderate distance therefrom. The velocity 
distribution in the i mmedi at e neighborhood is affected by all de-
tails of the different parts of the airplane and its exact compu-
tation is near to impossible. At a great distance, on the other 
hand, the air velocity caused by a passing airplane becomes negli-
gible . This appeals immediately to the common sense, and the re-
mark seems almost obvious and trivial . It is not, however, as 
the same docs not hold for the motion of the air in rear of the 
airplane. The wake of the airplane is maintained over quite con-
siderable distances , it can be neglected only at much greater 
distances than the disturbance created at the sides, and on top 
and bottom. 
An estimate of the downwash at the sides of the airplane in 
moderate distance will be obtained by substituting a monoplane of 
about equal span for the biplane cellule (if any) and by t~king 
into considcration the two- dimens ional flow caused by a lifting 
wing and discussed i n Reference 1, section III. It is s..l}own there 
that a two- dimensional flow is gradually established by the wings , 
~J .A.C·A. Techni ca l Note o. 216 3 
and that in the vertical plane through the wings the same config-
uration of flow , but of half of the f i nal r.;agn1 tud'3 in strength, 
will be found. For th e purpose o f thi s 1:Jote, j . t i s suf£ie;ieflt to 
consider only the main term of this flow, - whe~ cxpan~ ed into a 
Fourier ' s Series - having reference to an ell.~.pticill <i i st:rioution 
of the lift. This flow i s of vanishing intensity a t great dis-
tances from a s trai ght line, repre Genti ng the span. At the points 
of this span, the vertiGal veloci ty is cov='tant, sny, equc..l to u o ' 
Since t he appQ.rent mass of a st rai ght line of the It3nr:th 2a in 
a t wo-di mens ional flow iG equQ.1 to u2 n (that j s, I-rheD ;·.1ov i ng lat-
erally) and since the momentum of the fuml f~ov.' set up pcr uni t 
of time cquals the lift created, U o can bc compu ccd from the 
equat io n 
( 1) 
U o = 
This i s the finul downwash far behi nd the wing, According to the 
remark just mQ.de th e dO'Vffiwash at the points of t he \7i ng is half 
t hat magnitude . 
L (2) 
The p roblem is now reduccd t~ the investigatio n of the po t en-
tial flow around a stI'aigh'c 11:1C of the 1engt!.1 2a F.:OIT:L:'"lg l.J.Gcr-
ally wi t h the velo ci ty u 1 ir, c. pel.'.fc;:ct £ll].td ')t.herwis~ cL res t. 
The complex potential func:t,icn of '~his flm'l[ acco.:cl:l Ilgi:u equ8;c ion 
N.A . C· A. Technical Note No. 216 4 
38, of Reference 1, is 
l 
1 J ( 3) 
in which 
a = half span 
L = total lift 
v = veloe i ty of f lJ. gl::t 
~ = mass densit y of atr 
Cis a. constant, the l1'1.agni tude of which 
will be determi ned immediately. 
The velocity distribution i s then obtained from 
'r z ] F' = dF = i C 1 __ -;::=a:::=:::;;~==-
dz L a jGi'f3 a -- \ - 1 
al 
( 4) 
where only the last term in the bracket is variable. The verti cal 
velo city component is the imaginary part of (4), with reversed s i gn o 
At the points of the span, the second term of the span becomes 
i maginary and he nce, does not furnish a contribution to the down-
wash. At these points, equation (4) gives thus 
u = C 
a 
This constant downwash agrees with the one specified in equation ' 
(2), if C = L 
4 a ' n 
p 
V 2 
At the o ther pOints of the straight line 
containing the span , the vertical velocity component is dirocted up-
wards . Tho square root in (4) is then real, and. the upwash becomes 
- rll x ] 
u - C a - a J-x 2 - rl 
N.A .C .A. Technical Note No . 216 5 
where x denotes the distance f ro m the middle of the span. At 
x == co J u becomes zeto . 
At all other po ints of t he plane, ec;:ua '~io:} (4) furnishes the 
negative or positive downwash . Z = x + iy h·).s to 'oe inserted, 
and the expression (4) has to be redu.c e,l to tne form v + ui. 
Thi s can always be done easi ly by furJ.da ffi·:-r:"3u.l operations. The 
real and the imaginary parts give, then, the tVfO velocity compon-
ent s desi red . 
It is too cumber some , and unp r actical, to indicate the opera-
tion t o be performed in a fo rmula containing the trw space vn.ria-
bles x and y. It i s more convenien t to use an expansion of the 
downwash component , in a Fourier's Series, which rapidly converges 
at the region considered. I p roc eed to develop this Fourier 1 s 
Series. 
The variable term i n equation ( 4) is 
1 
2 -l/2 
Z (1 - (a'\ , 
a 
= 
a 
= 
Z t. L-
a) (~)2 _ r- (a 'f 1 / 1- a 
"-
Z/ 
( 5) 
2 -1/2 
Now the binominal expansion of (1 - (a \ gives Z/ I 
_1/2 
( 1 _ (a 2\ Z) ) 
2 (_1\( :-3 , 4 (_1 (_3\[5\ 6 
_. 1 + .1 (a 'i + \ \ 2 / ' .. 2 I (a '\ _ \ 2 /' 2 / \ 2.1 (a '\ 
2 \ Z j 1 x 2 Z) 1 x 2 x 3 \Zj 
., '1 6 8 63 J a 
=1+1(a G +
8
3(a +2... (fI.\+ 35 (a\+_,---(a +"(6) 
2 Zj \zj 16 ,Zj 128 Zj 256 \Z/ 
2 4 6 8 
= 1 + 0 . 5 r-a
z 
+ . 3750 (a + .3125 (a + . 2735(a . + ... 
' / z/ Zj \Z, 
N· A.G.A. Technica l No t e Jo . 216 6 
Introducing )OW , pO l a r coordinates 
XZ + yZ RZ x = cos r.p = -, R' 
tho real part of ( 6 ) becomes , (accordi ng to ~10ivrcl s f 0 r r.'lUlu ) 
1 + 0 . 5000 (ay R/ co s 2CP + 0 . 37 50 (~J4 COG 4CP + 
( 7) 
Hence the downwash becomes 
U :::: COG 2CP + 0 . 3750 (a\4 cos 4CP + 
\RJ 
8 
+ 0 . 3 125 ( a 6 RJ cos 6Cf) + 0 . 2735 (a \R/ co s 8 cP + .... ] 
Thi sis the d cs ir ed f or mula . 
The figure shows the meani ng of Rand cpo R is the distance of 
the point from the mi ddl e of t he span . cP is the angle betwcen 
the radiu s R (connect ing the mi ddle of the span and the point 
with the span}9 
(8 ) 
-./t.CcA. Tcchnical Note 0 . 216 7 
lJumeric ,J,l Exn.m.plcs 
Let the wei ght of an ai r pl ane be L = 3500 lb., and its span 
2a be 36 ft. The momentary velocity may be 80 mi./hr. The con-
ditions are so chosen as to give comparatively appreciable down-
v:ash. 
Consider first the po int in the horizontal of the wing, one 
span from the tip, for which R = 3a and ~ = O. The expression 
in front of the bracket is 
The bracket is 
=-L_~ = 3500 x 390 = 
4a2 TT V P 2 
4.2 mi. /hr. 
0.5 + 0.375 + 0 . 3125 + .. .. = 
7 3 4 36 0 , 055 +0.005 
+,) ,,000 
0 ~ 060 
Hence the negative dovmwash at the point is 0.06 x 4.2 = 0.2 5 
mi./hr. and the angle of dOVlnflash = 98~5 = 00003 = rd. 1/6 of 
one degree. Repeat the same compatation for a point in the verti-
cal middle plane of the airplane one span on top of the wing , 
that is, ~ = 900 and R = 2ao The conditions of flight mi ght 
be the same, giving again 0025 mi ./" r. as factor in front of the 
bracket. The latter now bccomes 
0~5 + 0.375 
z2 2~ 
-0.125 + Or024 
= -0.00 5 + O ~ C IC I 
- O. 1 ml. Tnr . 
N.A . C .A~ Technic~l Note No . 216 8 
Tho angl c of clownwash = - 0 8~ '''"' 710 of one dcgroe. 
It appears from these examples that the angle of downwash 
decreases rapidly wit h the distance f ro m the airplane. For prac-
tical cases it can probably al ways be computed exactly enough by 
using only the first term in t he bracket of the final result. 
Reference. 
1. Max . ..1 . Munk! El aments of the Wing Section Theory and of the 
Wing Theory . N. A. C.A. Technical Report No. 191. 
1924. • 


The velocity distribution caused by an airplane at the points of a vertical plane containing the span

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