It is argued that there should be an agreement as to what conventions to use in determining absolute coefficients used in aeronautics and in how to plot those coefficients. Of particular importance are the absolute coefficients of lift and drag. The author argues for the use of the German method over the kind in common use in the United States and England, and for the Continental over the usual American and British method of graphically representing the characteristics of an airfoil. The author notes that, on the whole, it appears that the use of natural absolute coefficients in a polar diagram is the logical method for presentation of airfoil characteristics, and that serious consideration should be given to the advisability of adopting this method in all countries, in order to advance uniformity and accuracy in the science of aeronautics

Munk, Max

English

NASA Technical Reports Server

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TECHNICAL NarES. 
NATIONAL ;~DVISORY COMMITTEE FOR AERONAU7ICS,. 
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No. 58 
AESOLUTE _Q.OE~FI CIEHTS A~m THE GRAPHICAL REPRESENTATION OF 
.AEROFOIL CHARAGTERIST les. 
By 
~1ax 1~UrLc, 
Natioca1 P<f'visor y Comrr.i ttee for Aeronautics . 
THIS oocw O'.r 0" L'M • FRO' THE FI 
!RON~UTles 
: 
1 ., 
.... ,.. ... '-f\.I:.r"'~ 
June, 1921 . 
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https://ntrs.nasa.gov/search.jsp?R=19930080835 2020-03-17T05:00:01+00:00Z
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NATIONAL ADVISORY COMlv:iITTEE FOR AERONAUTICS . 
TECHNICAL NOTE NO. 58. 
ABSOLUTE COEFFICIENTS AND THE GRAPHICAL REPP~SENTATION OF 
AEROFOIL CHARACTERISTICS: 
By 
Max MUllK.) 
National Advisory Committee for Aeronautics. 
It is customary to examine the aerodynamic qualities of 
an aerofoil by considering the coefficients of the forces, 
rather than the actual forces, corresponding to any particu-
1ar set of conditions. Such a coefficient, being always non-
dimensional (absolute), is the ratio of the actual force to 
sOme standard force corresponding to the given area, relative 
veloCity, and air density. It is only by the use of such 
• coefficients that the designer is able to judge the qualities 
of a profile and to compare, on the same b~sis, one profile 
vvith another. The reports of tests whether on models or in 
free-flight usually include both the Observed forc3s ~nd the 
calculated coeffiCients, although it is not uncommon to find 
in published reports only the coeffiCients. The angles of 
attack are always given to make these data complete. 
The use of absolute coefficients is almost universal in 
all sciences in all countries, which shows their advantage. 
Unfortunately in the science of aerona.uties there is SOme 
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lack of agreement as to conventions and there are not only sev-
eral kinds of coefficients in use in various countries but 
there are also differehces in the methods used in plotting them. 
In some countries there have been changes even in the standard 
both for the coefficients and for the methods of plotting. 
As a result, the oldar publications using obsolete Inethods are 
confusing to the average reader who is, perhaps, familiar with 
the current methods only. The fact that such changes have been 
made is sufficient proof that there are certain advantages or 
disadvantages connected with each scheme in use, since it is 
hardly likely that an entire country would change from one sys-
tem to another if all were of equal merit. Indeed, it may be 
shown that the lift and drag of an aerofoil supply an example 
of those quantities which require the use of a certain abso-
lute coefficient and a particular method of graphical represen-
tation, in order that the results may be interpreted fully . 
Coefficients . 
In aeronautics there are two types of absolute coeffic-
ients which demand particular attention. The first kind is in 
common use in the United States and England, the second kind 
in Germany. The essential difference lies in the "standard 
force" which is only half as great in the second kind as in 
the first. That is, the absolute coefficients of lift and drag 
are determined in the United States by the expression 
-where 
F = C P g 
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2 
S v .. .. . . . . . . . . . . 
F is the force 
p 
g the mass-density of the air 
S the area of the aerofoil 
V the relative velocity 
C the absolute coefficient of the force F. 
While the absolute coefficients of lift and drag are de-
termined in Germany by the expression 
F=CJ?. SV2 
2g 
It appears upon casual examination that the system in use 
in the United States is the more natural and therefore the bet-
tar on account of the omission of the coefficient 1/2. Upon 
a careful study, ~owever, it is evident that the second system 
is superior to the first in two respects. Since botl.1 sides of 
equations (1) and (2) represent a force, the expressions p 
g 
and ~ ~ must represent a force per unit area, that is, a 
2g 
pressure, and it is of especial importance to understand clear-
ly t:he exact pressure to which each refers. Otherwise it is 
not likely that the exact significance of the absolute coeffic-
ient s will be underst ood. The second expression 1 p"'(T2 . __ v ) 1S 
2 g 
the difference between the maximum pressure on the surface of 
a body due to air having a velocity V and the pressure in 
air at rest . This pressure difference is that given by the 
common Pi tot-tube and may be called the "Dynamical Pre ssure. II 
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In the German publications it is denoted by the symbol q. On 
the other hand the first expression P V2 has no physical 
g 
meaning and can only be understood and felt as HT wi ce the Dynam-
ical Pressure. II The t-IYO expressions are closely related to the 
expression for the kinetic energy of a moving body, 1/2 I .. N2 . 
The coefficient 1/2 resulting from the integration of VdV can-
not be avoided here . The expression IW 2 i s never considered. 
ural 
p 2 
-V g 
In forming absolute coe fficients the choice of the nat-
expression 1. P IF 
2 g 
instead of the meani~less expression 
as the standard pressure, not only gives the coefficient 
a definite r:hysical meaning, but also renders the quantities 
easily understood by enabling the density and the square of the 
velocity to be al ways grouped together and considered as the 
dynamical pressure. 
There are additional advantage s connecte d ~nth the use of 
the natural absolute coefficients based on the expres sion 1 
2 
£ V2 • The advantages are apparent wbe n it is necessary to make g 
use of certain theorems connecting 1 ift , drag and angle of at-
tack (see Technische Berichte 11-2). These theorems and the for-
mulae resulting from them are not only i nteresting from a phys-
ical point of vie w but are al so of great pract ical value to 
the designer of aircraft. Furthermore, these formulae are qu ite 
simple and it requires no more mathematical knowledge and cal-
culation to understand and to apply them than it requires to 
apply the simpl est formulae for the stresses in bent oeams . 
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These formulae demand the use of the natural absolute coeffic-
ients based on the actual dynamical pressure if the sin~ple form 
is to be retained. The use of the absolute coefficients which 
are no ~ standard in the United States and England> introduces 
additional factors confusing and likely to lead to error in sub-
st itut ion. 
Graphical Representation. 
There are t wo principal methods of representing graphic-
ally the characteristics of an aerofoil. In the United States 
and England it is customary to plot lift and drag coefficients 
as ordinates against angles of attack as abscissae, thus obtain-
ing two curves. The continental method, sometimes called the 
"polar diagram" * employs but a single curve in which the lift 
coeffiCients are plotted as ordinates and the drag coefficients 
as abscissae, The angles of attack are comrrlonly indicated on 
this diagram by figures along side of the curve. 
The results of a test on a model wing are plotted in Fi g . 
1, according to the usual American and British practice, and in 
Fig. ~, the same data are plotted according to Continental us-
age. These methods differ greatly and the true points of dif-
ference are not al ways well understood. In the first place the 
angle of attack has no definite significance aerodynamically, 
* If lift and drag are plotted in the same scale, the line of 
connection between any point of the curve and the origin of 
the system of coordinates is the vector of the force on the 
wing as to direction and size. Therefore, this diagram can 
be considered to be a polar diagram, the radii representing 
the absolute magnitude of the forces and the angle repre-
senting the angle between the force and the direction of mo-
tion. 
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for it is merely an agreement or convent ion which consider$ the 
chord as the direction of the section. Further, .the definition 
of "chord" is not clear in all cases. It fails .ent irely when a 
wing is twisted (vash-in or wash-out) or when the chords of a 
system of t wo or more wings are not parallel. Consequently, in 
plotting coefficients against angle of attack there is obtained 
no natural comparison between the characteristics of various aero-
fOils. The position of the Y-axis has no special physical mean-
ing and is unimportant for the qualities .of the aerofoil. Hence 
by using this method the designer Tenounces one of the advant-
ages - and the simplest too - wh ich are connected with plotting 
at all. 
The designer usually desires a large lift a nd a small dra~. 
These two quantities and their relation to each othar are most 
important in making an estimate of the value of an aerofoil. 
The angle of attack is merely a structural conside rat ion • . In 
order to obtain a connection between tl"e lift and drag when ·sep-
arate curves are plotted against angle of attack, it is necessary 
to carry through tedious mental processe s and the final result 
can not compare in vividness with the mental picture g iven by a 
glance at a polar diag=am. 
There are al so reasons why the polar diagram is the "natu-
ral" method of representing aerofoil characteristics graphically. 
Aerodynamical _theorens and actual tests prove t~at the lift de-
pends not 'upon the angle of attack but upon the flow about the 
,-.-~---~---~~ ---~~~-~------~--~~---------~ - ' "'~ 
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wing. 'i'hat is to say, the air flow around wings of the same 
sections but of different plan form is the same for equal li f t 
coefficients and not necessarily for equal angles of a t tack. 
Furthermore, the drag may be divi ded i:.1t 0 t wo part s, one of 
which depends upon the lift but neither of which depend up on 
the angle of attack. One of the furmulae previously mentioned 
(Technische Beriehte II-8) provides a very simpl e me t hod by 
which one may calculate that part of the drag w~ich is due only 
to the particular arrangement and proportions of the lifting 
surfaces. This part of the drag is independent of t he ae yofoil 
section and is called the "induced drag . 11 The induced drag may 
be considered as the minimum limit of drag consistent with the 
aspect ratio used and is an ideal which may be approached through 
the reduction of "section drag" but which can never be equaled. 
This "section drag ll is conditioned by the a erofoil section and 
must be obtained from tests either on models or in , f ree flight. 
This part of the drag is determi ne d for exampl e , by the change 
in the performance of an airplane when only the total load is 
changed. The first and sometime s the more i mportant part of the 
drag may be calculat ed very quickly with a slide r u l e , and wit h-
out the necessity of t e sts, may be plotte d as a parabola, depend-
ent upon the lift. The formula for this llinduced" drag is: 
D = 1 L2 . 1 It v2 J3' B2 
!·3 
where L is the lift; B the spa n; and. V2 P;8 the dynamical 
-------
--.--~--
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pressure. Written in absolute coefficients, defined by 
P :3 
D == Dc 2 V S) et c. 
·,vhere S is the area, the same formula "becomes 
1 D --c - \T 
3 1:3 2 •••. . (la) 
This formula holQS for single aerofoils; and for systems of aero-
foils, B is to be replaced by kB w~ere k is a coefficient 
.someivhat different from 1. This fOTIiiula represents a para,bola 
:vhich however, cannot be plott e d deper.dent upon the angle of at-
tack, Without tests, because there is no definite relation be -
tWeen the lift and the angle of attack. The de s igner who uses 
the plots of lift and drag against angle of attack instead of the 
polar diagram gives up half of the advantages to be obtained from 
the use of the formulae. 
Regardless of the attitude of the designer to vmrds the meth-
od which he uses to plot aerofoil characteristics, it is certain 
that his conclusions are influenced by ~hese diagrams, and that 
an unfavorable or obscure diagram may lead to a wrong conclusion. 
The curves of lift ~lotted against angle cert a inly do this very 
often. For instance, a designer may be led to co::npare t rIO dif-
ferent wings, or even t ;vo different sections, at the same angle 
of attack instead of at the same lift or drag coefficient. An 
ingenious man usually draws correct conclusions; but it is an ad-
vantage to use . diagrams which may be also interpreted by men WJ.l0 
are not specialists in aerodynamics. 
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other Pos3itilities of Grafhical Ren~esentati9n. 
The principal difference bet ;;~Ben the ~ ~o kinds of plo-ctin6 
ment ioned is the change of the variables. There are spec i3.1 ad-
vantages connected with the plotting of the lift and drag coef-
ficients directly against each other. Now ttese advantage s would 
not vanish if, instead of plotting the coefficients themselves. 
funct ions of them were taken. It is worth while to compare the 
advantages of several such diagrams. 
Any two such diagrams are mathemat ically connecte d. with each 
other. Any construction in the one diagram can be repeated in 
the other, and to each curve dr'Q.vvn in the one belongs a corres-
ponding curve in the other, In general, the corresponding curve 
is not a straight line if the original curve is a straight line. 
The chief difference between different diagrams lies in the type 
of curve by which the most important relations are represented. 
In the diagram generally used, Lc against Dc the curves of 
constant Lc and Dc of constant L/D, and of constant velocity are 
straight, anci the 11 induced" coefficient curve and the important 
Dc curves for constant power = const. are curved lines. 
L 3/2 
c 
If one coordinate is the drag coefficient itself, the addition of 
a constant drag coeffiCient, for instance, when proceeding from 
the w~ngs to the entire airplane, can be represented by merely 
transferring the origin of the system of coordinates. The origi-
nal curve remains unaltered, This quality of the diagram is so 
useful that a diagram without it would be inferior. Whence it 
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follows that Dc should. always be plctteci di::.-ectly in one direc-
t ion . To so choose coordinate s t rlat tne CUl'ves for constant L(u 
are straig:J.t is in &. sinaller deg::-ee ajvantageous. It is true 
-chGi.li LID is irequently consider9d in p!'esent practice 1 but this 
i s done) not becau.se it means very much, 'out beca·l1.se in the pres-
ent diagrams this quantity is the only one givi~g a direct rela-
t ion bet ween 1c and Dc. It rrould be better if the curve s for 
aonstant po Ner are straight, for the power is more i!!lportant than 
the angle of gliding. This can be obtained by plotting LC 3/2 in-
stead of 1c against Dc' The induced coefficient rem4ins Q curved 
l~ne, and all advantages of t~e first diagram remain too. It is 
not even necessary to calculate and to put in the values of tha 
1 . 5 power of 1C ) for, as in logarith~ic diagrams, it is quite suf-
ficient to use a proper variable graduation of ~he corresponding 
~~is of coordinates . 
Another possibility would be the plotting of Dc3/2/Lc against 
Dc . The power would be plotted as it were against the drag, where-
as in the preceding diagram it can be considered as being plotted 
against the lift . This seems to be more natural. 
LID against Lc sometimes used in England, gives straight lines 
:6r constant DC) but the addition of a constant Dc requires a new 
curve; nor are lihe curves for the induced coefficient or for con-
stant powe r nor for Lc straight. The drawing of a new curve when 
adding a constant Dc is still more complicated than before . 
There remains therefore only the diagram 10 3 / 2 against Dc 
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as a competitor to the diagram Le Ggainst De. The diff erences 
be1; '.7een the t'.70 diagrams are not consid.erable. It is convanient 
t o have st raight lines for constant po wer but the odd l.)O i~·er of 
LCe is sometimes confusing. In any case the advanta,ges a re n ot 
sufficient to compensate for the di sadvar:tage of us ing c_ia gI'E..ws 
differing from those used in most other countries. 
Conclusions. 
In addition to the i~portant features connected with the 
use of natural aosolute coefficients in polar diagrams there are 
several minor advantages. A few of the special a ppli·cations are 
given in the above references. ~1 the whole it apFears that the 
use of natural absolute coefficients in a polar diagram is the 
logical method for presentat ion of a e rofoil characteri s t ics . 
Serious consideration should be given to the advisability of 
adopting this rr:ethod in all countries. The actual adoption would 
be a great advancement of uniformity and accuracy in the science 
of aeronaut ics. 

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Absolute Coefficients and the Graphical Representation of Airfoil Characteristics

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