The supercritical profile has an upper surface whose curvature has two minima, one in front of the mid-chord station, the other one rather close to the trailing edge and having a lower value than the first one. The curvature of the lower surface remains larger beyond the mid-chord station, changes sign and increases to a pronounced maximum not far from the second curvature minimum of the upper surface, and drops to zero at the trailing edge.

BACKGROUND OF THE INVENTION 
The present invention relates to a supercritical air-foil, particularly a 
wing profile for an aircraft, or a related type air-foil having a specific 
design Mach number not exceeding about 0.85 and having a maximum 
thickness, in units of chord length of d/l=0.13.+-.20%. 
Supercritical wing profiles are known in a variety of profile contours and 
shapes resulting accordingly in different pressure distributions along 
upper and lower surfaces of the air-foil. Basically, two lines of 
development have resulted here, one of which is represented by German 
printed patent application No. 2,254,888 (and also U.S. Pat. No. 
3,952,971). The profiles as disclosed in this patent develop approximately 
constant pressure along the upper surface because the supersonic speed 
zone along the upper surface covers a rather large portion of the profile 
depth. These profiles have also be termed rooftop profiles; they do indeed 
produce an increase in lift but they have the disadvantage of an 
increasing drag resistance of the profile due to the rather abrupt 
pressure increase in more rearwardly located areas of the profile. 
Moreover, it was found that these particular profiles, being optimized for 
a particular design case, exhibit significant disadvantages and drawbacks 
for off-design operating conditions. For example, these profiles have the 
tendency that for lower than the design Mach number two shocks are 
produced which increase the drag resistance significantly. On the other 
hand, in the case of an increase in the Mach number, at which the profile 
is operated, the supersonic speed zone is bounded in the rear portion of 
the profile by shock which becomes rapidly stronger with an increases in 
speed and leads directly to an increase in resistance and to an increase 
in the pressure gradient, which in turn endangers increasingly the 
boundary layers as far as its separation is concerned due to the shockwave 
that is produced. 
The second line of development is particularly derivable from the two 
German printed patent applications Nos. 2,608,414 and 2,626,279 (see also 
U.S. Pat. No. 4,072,282). The profiles disclosed in these patents are also 
known as the Peaky profiles. The pressure distribution for these profiles 
is particularly characterized by a suction peak in the nose region and 
have, therefore, rather favorable resistance values for the design 
parameters. On the other hand, this particular type of profile has a 
drawback in that the pressure distribution is rather sensitive to changes 
in the angle of attack as well as to changes in the Mach number. The rear 
boundary of the supersonic zone sonic line terminates in these off-design 
operating cases in a rather rapidly increasing shock at shifting 
positions. The minimum lift as produced is prematurely limited in that the 
rather rapid increase in resistance introduces a non-linear lift moment 
which is the result of a shock-induced boundary layer separation. 
Therefore, an aircraft having such a wing experiences significant changes 
in pitching moment if the speed deviates significantly from the design 
speed and/or if the angle of attack deviates from the design value. 
Accordingly, the manouverability of such a craft suffers due to the 
sensitivity of the flight parameters to these external interferences and 
changes in operating conditions. For lower speeds the Peaky profile is 
endangered as far as lift is concerned by laminar nose separation. 
DESCRIPTION OF THE INVENTION 
It is an object of the present invention to avoid the drawbacks of the 
air-foil profiles outlined above and to proide a new and improved air-foil 
profile which combines the advantages of the two known profile types. 
Therefore, it is a particular object of the present invention to retain 
the favorable drag resistance characteristics of the Peaky profile and to 
combine this advantage with the rather favorable lift properties of the 
rooftop type profile. 
It is another project of the present invention to provide a profile which 
is to a lesser extent limited by the design parameters but whose behavior 
under off-design conditions still results in favorable performance. 
It is a specific object of the present invention to provide a low drag 
resistance, high lift producing profile which will not produce shocks, or 
only insignificant ones over a large range of Mach numbers and for a large 
range of different angles of attack. Moreover, in the case of a still 
increasing Mach number, the inevitably occuring shock should be stabilized 
in its position in a particular range of the profile depth. 
In accordance with the prefered embodiment of the present invention, it is 
therefore suggested to provide an air-foil profile in which the maximum 
upper surface thickness is located at a distance of 0.42.+-.10% (in units 
of chord length) from the leading edge of the profile; the corresponding 
location of the maximum thickness of the lower side is located at a point 
spaced from the leading edge by 0.3 (in units of chord length) +25%, -10%. 
The curvature defined as the inverse value of the radius of curvature of 
the upper surface is between 0.3 and 0.4 for a profile depth from 0.3 to 
0.55; the curvature of the upper surface is below 0.4 for profile depth 
from 0.55 to 0.7; for a profile depth of about 0.9 the curvature of the 
upper side should be approximately 0.2 and at a profile depth within the 
range from 0.5 to 0.6, the curvature has a point of inflection. 
Preferably, the curvature of the lower surface is larger than the 
curvature of the upper surface for chord with distances or profile depth 
from the leading edge up to about 0.55. The curvature changes sign at 
about 0.6 to 0.7 profile depth, increases to a maximum of about 1.3 and 
drops to zero at the trailing edge. 
An air-foil having these features produces a pressure distribution having a 
rather significantly pronounced suction range in the front portion of the 
profile which drops continuously i.e. shock free for the rated design case 
and at a relatively small pressure gradient towards the trailing end of 
the profile. In off-design operating ranges, such as changes in the angle 
of attack, shocks do occur on the upper surface of the profile for rather 
large angles of attack. However, these shocks are localized and stabilized 
in a very narrow range as far as profile depth is concerned. Accordingly, 
undesired changes in moment are avoided, and the performance of the craft 
remains favorable. An increase in Mach numbers will also result in shocks 
on the upper surface, but again, they are limited to a very narrow range 
as far as profile depth is concerned and do not produce significant 
changes in moment. This particular way of stabilizing the shock producing 
range rather far in front of the trailing edge avoids separation of the 
flow rearward from the shock; this is true for a wide range of angles of 
attack and for a wide range of Mach numbers. The inventive profile has 
favorable drag resistance coefficients at favorable lift coefficients, 
favorable off-design properties and favorable performance characteristics 
which render it particularly applicable to modern commercial aircrafts.

Proceding now to the detailed description of the drawing, FIG. 1 shows the 
shape of the air-foil profile in accordance with the prefered embodiment 
of the invention. The contour is plotted in a relation to a superimposed 
coordinate system, x/l and z/l, wherein l is the chord length and the 
chord line is situated on the abscissa. Thus, the relative profile depth 
has been plotted along the abscissa, and the relative profile thickness 
z/l has been plotted along the ordinate of the coordinate system. 
Thickness values are therefore distinguished for the upper and the lower 
surface, the dividing line being the chord line. All depth values for the 
profile are taken from the leading edge which is the zero point for the 
x/l values. 
This particular profile has certain features which are similar to features 
of known air-foil profiles. Such features are, a rather flat upper surface 
and a lesser flat lower surface having a concavely shaped portion more to 
the rear of the profile. It should be emphasized that FIG. 1 as well as 
other Figures are, in fact, drawn to scale, but the decisive criteria and 
critical elements of the shape in accordance with the invention are very 
difficult to be drawn and are, therefore, not well recognizable. The 
criterias outlined above and being responsible for the desired pressure 
distribution can be deduced from the additional figures. The tables of 
FIGS. 18 and 19 define the contour of such a profile and by direct x and z 
values. 
Turning now to FIG. 2, two diagrams are plotted here, the dash-dash line 
depicts profile thickness Z.sub.D /l against the relative profile depth 
x/l, resulting in a so called dropshaped contour. One can see from this 
particular dashed curve that the maximum thickness of the profile is 
located between 0.3 and 0.4 of the relative profile depth x/l taken from 
the front. This particular dimension and contour has to be differentiated 
on one hand from the rearward position of the maximum thickness of the 
upper surface and from the corresponding disposition of the maximum 
thickness of the lower surface. As stated above, the thickness of the 
upper surface and the thickness of the lower surface are z/l values and 
distances of surface points from the chord line. The maximum thickness 
values and their locations are shown in the box underneath FIG. 1. 
Accordingly, the maximum thickness of the upper portion of the wing is 
located at a relative chord distance (x/l).sub.mo =0.42 while the maximum 
thickness of the underside is located at (x/l).sub.mu =0.3. The mean 
chamber line also plotted in FIG. 2 being the center line between upper 
and lower surface profile contours and runs, therefore, in parts a little 
below the abscissa axis including particularly the nose portion. 
Decisive for the pressure distribution along upper and lower surfaces of 
the profile is the curvature distribution of upper and lower surfaces. 
This curvature distribution is plotted in FIG. 3 whereby in particular 
"curvature" is defined as the inverse of the radius of curvature in any 
point along the upper and lower surface and in a plane that includes a 
chord line in full. 
The curves in FIG. 3 are plotted to scale and they show that the upper side 
is, in fact, exclusively concavely shaped as far as the geometry in the 
particular coordinate system of FIG. 1 is concerned. In reality, of 
course, the profile is convex throughout the upper surface and most of the 
lower surface. The curvature is particularly characterized by two 
curvature minima (in absolute values), one being in about 40% relative 
profile depth and another minimum of curvature is located at about 90% of 
the relative profile depth. One can also say that these two curvature 
minimums of the upper profile surface are at 0.4 and 0.9 chord stations 
from the leading edge. In between these two minima is located a curvature 
maximum at about 65% profile depth, i.e. a 0.65 chord station from the 
leading edge. Within the definition given above, the first minimum at the 
0.4 depth or chord section has a value of about 0.3, the maximum in 
between at the 0.65 chord station is about 0.4, and the second minimum at 
the 90% depth point or 0.9 chord station is about 0.2, all curvature 
values as absolute values. 
The radius of curvature of the upper surface is therefore not constant for 
a significant range of depth values and chord stations as is the case in 
the prior art profiles; in fact, the radius of curvature of the novel 
profile's surface varies continuously. Beginning with the small nose or 
leading edge radius (large curvature) the curvature reduces steadily to 
the minimum at the 0.4 chord station, increases again slightly to a rather 
shallow maximum at about the 0.65 chord station from which maximum the 
curvature decreases and about the 90% point or 0.9 chord station the 
profile has actually an absolute minimum i.e. it is the flattest. In 
addition the curvature changes direction (point of inversion) at about 55% 
profile depth; the location of this point is of great significance as far 
as the design pressure distribution and the shock location stabilization 
is concerned. 
As far as the curvature of the lower surface is concerned, up to about 55% 
relative depth the curvature is almost consistently larger (absolute) than 
on the upper surface. Behind the 0.55 chord station the curvature reduces 
below corresponding values (absolute) on the upper surface (i.e. the lower 
surface is flatter here). At about 64% depth the curvature is 0 and 
reverses, that is to say, beyond a 0.64 chord station the lower surface 
becomes actually concave whereas it was convex for about two thirds back 
from the leading edge. However, it should be mentioned that the passage 
through zero of the curvature is not a point of inversion of the function 
K=f(x/l) in the mathematical sense but one usually designates this zero 
point as a curvature change point because the contour proper changes here 
from convex to concave. Beyond this zero crossing the curvature reaches a 
maximum of about 1.3, at about 85% depth and the curvature drops to zero 
at the trailing edge. 
After having described the relevant parameters of the profile we now turn 
to FIG. 4 showing the pressure distribution as it results directly from 
the curvature as described under consideration of additional aerodynamic 
parameters. This pressure distribution is plotted for a design case, i.e. 
a design Mach number of 0.75 which is a normal cruising speed. In 
accordance with this diagram the under pressure drops rapidly in the 
leading edge portion and remains approximately constant in the range from 
about 20 to 30% depth from the leading edge while dropping continuously 
towards the trailing edge. This particular, continuously shock-free rise 
in pressure is very favorable as far as the boundry layer is concerned 
because it avoids boundary layer separation which, of course, would 
constitute a loss in lift and an increase in drag resistance. The lower 
surface likewise experiences a rapid rise in underpressure in the more 
forward portion of the profile, reaching a maximum at the depth of about 
30%, dropping to zero at about 60% and resulting in an excess pressure 
along the underside, from the 0.6 chord station back to the trailing edge 
and particularly in accordance with the concave contour of the profile. 
The pressure remains consistently higher on the lower surface than along 
the upper surface. 
The FIGS. 5 through 12 show characteristics and measuring results of the 
invention profile, particularly for so called off-design conditions i.e. 
for cases in which the plane does not move in accordance with the design 
Mach number and/or does not operate at the design angle of attack. Each of 
the figures and graphs is accompanied by a box identifying curve symbols 
and associating them with the relative parameters. Herein M.infin. denotes 
the Mach design number; .alpha. is the angle of attack in degrees, Ca is 
the effective lift coefficient, C.sub.w is the effective drag resistance 
coefficient, C.sub.m is the pitching moment; and Re is the Reynolds 
number. The FIGS. 5 through 9, as stated, show how the angle of attack 
modifies the pressure distribution assuming that the Mach number has the 
design value of 0.75. 
FIG. 5, particularly, shows the pressure distribution for smaller angles of 
attack, and one can readily see that the pressure distribution differs 
very little from the pressure distribution of FIG. 4 in which, of course, 
the air-foil has the rated and design angle of attack. There is, however, 
a slight dent in the pressure curve in the range between 0.2 and 0.4 chord 
stations, however, that dent still does not produce any shockwave. 
FIG. 6 shows the pressure distribution for larger angles of attack .alpha., 
whereby the supersonic zone is in fact bounded in its rear by a 
compression shock in the range being developed at a location of the upper 
surface rearwardly displaced from the leading edge by about 50 to 60% 
profile depth. These compression shocks for angles of attack from 2.280 
and 2.690 are rather closely positioned. FIG. 7 shows the pressure 
distribution for still larger angles of attack, and one can readily see 
here that for all of the three different angles of attack the disposition 
of the compression shocks is narrowed as far as the profile depth is 
concerned; the shocks are produced in all cases around the 0.6 chord 
station. This aerodynamic phenomenon is a characteristic feature of the 
profile in accordance with the invention. Higher angles of attack 
inevitably produce compression shocks, but they are all localized as far 
as their position is concerned, in a profile depth range from about 10 to 
20% around the 0.6 chord station. This way the shocks are stabilized well 
in front of the trailing edgs, so that a sufficiently low pressure 
gradient develops over a rather large depth range, and boundry layer 
separation and, therefore, premature drag increases and lift-collapse is 
avoided. Localization of the shock producing portion as well as the 
avoiding of premature trailing edge separation indeed leads to moment 
coefficients which vary very little which feature in turn is a very 
favorable aspect of the profile as far as flight performance is concerned. 
FIG. 8 illustrates the pressure distribution for negative angles of attack; 
the flow along the underside will become supercritical but remains shock 
free and does not even separate in the rear portion of strong profile 
curvature. 
FIGS. 9 and 10 show the relationship between, on the one hand, the design 
Mach (M75)number and lower, actual Mach numbers, at slightly different 
angles of attack, and the pressure distribution, on the other hand. FIG. 9 
has validity for Mach numbers below the design Mach number and FIG. 10 
shows the pressure distribution for larger Mach numbers. For the smaller 
Mach numbers as per FIG. 9, one will observe a small drop in the pressure 
followed by a slight increase at about the 0.4 chord station, but that 
pressure drop is insignificant and will rapidly be "filled" for increasing 
Mach number so that there will be no shock. In the case of Mach numbers 
above the design Mach number as shown in FIG. 10, compression shocks are 
observed on the upper surface in the range between the 0.6 and 0.7 chord 
station. Again, it can be said that these shocks are stabilized as to 
their position, i.e. they are limited to a very narrow range on the upper 
surface of the wing. These measurements have shown also here that the 
off-design operating conditions of the profile do in fact lead to 
favorable flight performance. 
The FIGS. 11 and 12 show the relationship between angle of attack .alpha. 
plotted along the abscissa, Mach number (parameter of the family of 
curves) and lift, lift coefficient C.sub.a. The family of curves shown in 
FIG. 11 has validity for Mach numbers as a parameter below the design Mach 
number. FIG. 12 shows a family of curves in which the Mach number 
parameter is above the design Mach number. Both figures show a rather high 
maximum lift. 
FIGS. 13 and 14 show the relationship between lift coefficient Ca and 
pitching moment represented here by the coefficient cm. The Mach number is 
again the parameter for the family of curves. Accordingly, the 
non-linearity of the moment is rather weakly developed in spite of a 
rather strong rear profile load also called rear-loading. This holds true 
for Mach numbers which cover the required off-design range for this 
particular case and Mach design. 
FIGS. 15 and 16 show drag resistance vs. lift in dependency upon the Mach 
number, whereby again FIG. 15 depicts a family of curves well below design 
Mach numbers as parameter and FIG. 16 is analogously plotted for above 
design Mach numbers. Finally, FIG. 17 shows drag coefficient C.sub.w vs. 
Mach nembers, the lift coefficient C.sub.a being the parameter in this 
case. 
All these curves demonstrate most vividly the effect of the pressure 
distribution that results from the inventive profile. The range of low 
profile drag resistances has been considerably expanded consistently with 
a lack of sensitivity of the pressure distribution to variations in actual 
Mach number and angle of attack. The pressure distribution is indeed 
favorable as far as drag resistance is concerned. The resistance level of 
cw=0.01 will be exceeded only for lift values which are well above the 
design case. 
FIGS. 18 and 19 show respectively table 1 and table 2 in which x values, 
i.e. true profile depth values are associated with Z or thickness values 
for a profile, whereby table 1 shows the Z values for an upper surface and 
table 2 shows the Z values from a lower surface, the chord line being Z=0 
in either case. 
The geometry of the profile in accordance with the invention produces a 
design pressure distribution which combines a large lift with a low 
resistance. The advantage of this particular pressure distribution is 
retained over a considerable range in off-design operating parameters. The 
particular pressure distribution reacts only very little against external 
disturbances such as changes in the Mach number, changes in the angle of 
attack or against local construction deviation such as transition to flaps 
or elastic deformations of the air-foil. These properties of the inventive 
profile fullfills the essential requirements for introducing the 
supercritical wing into the modern aircraft design development and 
construction. 
It can readily be seen that profiles of the type can be used as propellers, 
rotors, rotor blades in fluid machines or the like. The invention is not 
limited to the embodiments described above but all changes and 
modifications thereof not constituting departures from the spirit and 
scope of the invention are intended to be included.