The method has been used to design a series of low speed airfoils, many of which have been successfully applied as listed below.

To put the method in some perspective, it helpful to briefly outline the two general approaches to airfoil design. There are "direct" and "inverse" methods. Direct methods are based on the design-by-analysis approach in which the airfoil shape is specified first. The given airfoil is then analyzed to determine, say, its velocity distribution. The airfoil shape is then adjusted until the desired velocity distribution is obtained.

One difficulty with direct methods is that the designer spends a great deal of time focusing on the velocity distribution; the geometry is secondary. Thus, it would be more advantageous to specify the velocity distribution from the outset and from that determine the airfoil shape. Such methods are called inverse methods.

PROFOIL-WWW is based on an inverse method as described. The velocity distribution is specified by a limited number of parameters. In fact, at a minimum only 8 parameters are needed to define the entire velocity distribution (see screen grab below). In the method, such an airfoil is termed a four-segment airfoil. Of course, for more complex airfoils (more practical designs) more parameters are needed. But one advantage of the method is that the entire velocity distribution about the airfoil does not have to be described point-by-point, which could be quite tedious, depending on how it is implemented.

The method has it roots in a branch of mathematics called conformal mapping. Without going into any details, it is only necessary to know that the airfoil is generated from a circle that is "mapped" into an airfoil as depicted in the figure above.

The Joukowski airfoil problem is the simplest application of conformal mapping to airfoil design. The mathematical transformation is fixed and the airfoil is moved to generate different airfoil geometries. An alternative approach is to fix the geometry and change the transformation to get different airfoils. This strategy is used in the current approach. The transformation, however, is not changed explicitly by the designer, but instead determined from the parameters that define the velocity distribution. Thus, conformal mapping is merely used as a bridge to allow the designer to specify the velocity distribution. The details of the method are for the most part transparent to the user.

Another aspect of the method is that "multipoint" design can be performed. The velocity distributions previously shown were for a single angle of attack. When the velocity distribution is specified for a single condition it is refered to as "single point" design. Typically, however, good performance is required over a range of angles of attack, say, two different angles of attack. For example, high-lift (low speed/high angle of attack) performance may be required as well as low-lift (high speed/low angle of attack). The process of prescribing the velocity at two (or more conditions) is referred to as "multipoint" design.

PROFOIL has a multipoint design capability. For instance, the velocity distribution can be prescribed for the upper surface at a high angle of attack while simultaneously the velocity distribution can be prescribed for the lower surface at a low angle of attack. This multipoint design feature will be discussed in more detail in the examples.

PROFOIL-WWW is setup to allow web users to design airfoils using the html forms support. The parameters are specified and submitted. Calculations are performed on the server and the associated figures and results are written to html files on the fly and displayed.

The tutorial begins with a simple four-segment airfoil. The various design parameters are introduced before moving on to a six-segment airfoil. Finally, a special form allows the user to prescribe any number of segments so that practical airfoils can be designed. In this regard, several airfoil templates are given based on existing designs, eg, the SD7037 airfoil and others.

The Web version of the code allows for the prescription of the inviscid velocity distribution about the airfoil. The full-featured version of the code has additional capabilities that are not available on the Web. In particular, a boundary-layer method is included to permit the specification of boundary-layer developments. For instance, the development of the boundary-layer shape parameter can be prescribed on the upper surface to allow for maximum lift or extended runs of laminar flow, depending on the design requirements. The method also permits inverse design with flaps, specification of geometric constraints (thickness, camber, etc.). More details of the method can be found in the references below.

- Selig, M.S. and Maughmer, M.D., "Multipoint Inverse Airfoil Design
Method Based on Conformal Mapping,"
*AIAA Journal,*Vol. 30, No. 5, May 1992, pp. 1162-1170. - Selig, M.S. and Maughmer, M.D., "Generalized Multipoint Inverse
Airfoil Design,"
*AIAA Journal,*Vol. 30, No. 11, November 1992, pp. 2618-2625.

- Selig, M.S., "Multipoint Inverse Design of an Infinite Cascade of
Airfoils,"
*AIAA Journal,*Vol. 32, No. 4, April 1994, pp. 774-782. - Saeed, F. and Selig, M.S., "A Multipoint Inverse Design Method for Slot-Suction Airfoils," AIAA Paper 95-1857, AIAA 13th Applied Aerodynamics Conference, San Diego, CA, June 1995.

- Selig, M.S. and Gopalarathnam, A., "A Multipoint Inverse Method for Multi-Element Airfoil Design," to be presented, AIAA 14th Applied Aerodynamics Conference, June 18-20, 1996.

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