Airfoil has least amount of drag under given restrictions of boundary layer transition position, lift coefficient, thickness ratio, and Reynolds number based on airfoil chord. It is suitable for use as wing and propeller aircraft sections operating at subsonic speeds and for hydrofoil sections and blades for fans, compressors, turbines, and windmills

Chen, A. B.

Peterson, J. B.

English

NASA Technical Reports Server

October 1975	 B75-10256 
NASA TECH BRIEF 
Langley Research Center 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Design Procedure for Low-Drag Subsonic Airfoils 
. 
.
A procedure has been developed for designing an 
airfoil for use at subsonic speeds. The airfoil has the 
least amount of drag under the given restrictions of: 
Boundary-layer transition position 
Lift coefficient 
Thickness ratio 
Reynolds number based on the airfoil chord. 
Low-drag subsonic airfoils are not only suitable for 
use as wing and propeller aircraft sections operating 
at subsonic speeds but also for hydrofoil sections for 
boats and blades for fans, compressors, turbines, and 
windmills. 
The low-drag subsonic airfoil design procedure 
consists of four steps: 
1. Development of a Formula To Calculate Airfoil 
Drag— The formula to determine this is derived from 
Betz' formula for the calculation of airfoil drag from 
surveys of the wake behind the wing. By suitable 
assumptions, such as static pressure being constant 
through the boundary layer and velocity in inviscid 
flow being constant across the boundary layer, Betz' 
formula can be expressed in terms of the boundary-
layer momentum thickness 0 and shape factor H so 
that:
20 Ud 
Cd = 7 j ()2 + H [e-) 2 ()1 
where Cd = drag coefficient of the airfoil, 
o = momentum thickness of the boundary 
layer at the trailing edge, 
c = chord of the airfoil, 
Ud	 velocity at the edge of the boundary layer 
at the trailing edge of the airfoil, 
u 00 = velocity in the free stream, 
H = 6*10 = shape factor of the boundary 
layer at the trailing edge, and 
= displacement thickness of the boundary 
layer at the trailing edge.
2. Calculation of Optimum Pressure Distribution - 
Using the formula for drag coefficient and a suitable 
boundary-layer computation method, the drag of the 
airfoil can be calculated if the Reynolds number is 
known. Also, the lift coefficient on one side of the 
airfoil can be closely estimated if the pressure 
distribution on that side is known. Not every pressure 
distribution can be calculated, but a computer 
program has been developed which explores a great 
many in a small amount of time. The printout 
permits quick determination of the range of the 
constants which lead to lower drags and exploration of 
these areas in greater detail. 
3. Calculation of Airfoil Shape To Conform to 
Optimum Pressure Distribution - After the optimum 
pressure distribution is obtained for a given lift 
coefficient, this pressure distribution is used in 
another computer program which develops the 
geometry of the airfoil. This program calculates the 
pressure distribution on the surface of a certain airfoil 
at a given angle of attack and systematically modifies 
the airfoil geometry until the desired pressure 
distribution is obtained. The input to the program is 
an initial geometry, the direction of the free stream, 
and the specified pressure distribution. 
4. Calculation of Airfoil Performance in Viscous Flow 
- When the airfoil configuration is obtained from the 
design program, use is made of a two-dimensional 
viscous-flow computer program to examine the 
performance of the airfoil in different modes of 
operation. Computed results for an airfoil optimized 
for minimum drag at a lift coefficient of 0.9 reveal 
that, at the climb lift coefficient, the drag is less than 
for conventional airfoils currently in use on many 
popular general-aviation aircraft. 
This design procedure will provide different airfoil 
configurations which generate the same optimum 
pressure distribution, because the optimum pressure 
(continued overleaf) 
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distribution exists only on the upper surface of the 
airfoil and no pressure distribution is specified on the 
lower surface •during the iterative procedure. Dif-
ferent thickness distributions and/or maximum 
thickness/chord ratio may be incorporated in the 
initial geometry. Based on this nonuniqueness, a 
series of airfoils may be developed which will have a 
wide range of applications. To broaden the spectra 
even further, the optimization program may be 
applied to pressure distributions which result in 
different lift coefficients. This will lead to families of 
airfoils which have the minimum amount of drag for 
various lift coefficients.
Note: 
Requests for further information may be directed 
to:
Technology Utilization Officer 
Langley Research Center 
Mail Stop 139-A 
Hampton, Virginia 23665 
Reference: B75-10256 
Patent status: 
NASA has decided not to apply for a patent. 
Source: John B. Peterson, Jr.

Langley Research Center and

Allen W. Chen of

National Research Council

(LAR-11351) 
Categories: 03 (Physical Sciences) 
06 (Mechanics) 
09 (Mathematics and 
B75-10256	 Information Sciences)


Design procedure for low-drag subsonic airfoils

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