The direct airfoil design methods involve the specification of a section geometry and the calculation of pressures and performance. One evaluates the given shape and then modifies the shape to improve the performance.
The two main subproblems in this type of method are
The simplest form of direct airfoil design involves starting with an assumed airfoil shape (such as a NACA airfoil), determining the characteristic of this section that is most problemsome, and fixing this problem. This process of fixing the most obvious problems with a given airfoil is repeated until there is no major problem with the section. The design of such airfoils, does not require a specific definition of a scalar objective function, but it does require some expertise to identify the potential problems and often considerable expertise to fix them. Let's look at a simple (but real life!) example.
A company is in the business of building rigid wing hang gliders and because of the low speed requirements, they decide to use a version of one of Bob Liebeck's very high lift airfoils. Here is the pressure distribution at a lift coefficient of 1.4. Note that only a small amount of trailing edge separation is predicted. Actually, the airfoil works quite well, achieving a Clmax of almost 1.9 at a Reynolds number of one million.
This glider was actually built and flown. It, in fact, won the 1989 U.S. National Championships. But it had terrible high speed performance. At lower lift coefficients the wing seemed to fall out of the sky. The plot below shows the pressure distribution at a Cl of 0.6. The pressure peak on the lower surface causes separation and severely limits the maximum speed. This is not too hard to fix.
By reducing the lower surface "bump" near the leading edge and increasing the lower surface thickness aft of the bump, the pressure peak at low Cl is easily removed. The lower surface flow is now attached, and remains attached down to a Cl of about 0.2. We must check to see that we have not hurt the Clmax too much.
Here is the new section at the original design condition (still less than Clmax). The modification of the lower surface has not done much to the upper surface pressure peak here and the Clmax turns out to be changed very little. This section is a much better match for the application and demonstrates how effective small modifications to existing sections can be. The new version of the glider did not use this section, but one that was designed from scratch with lower drag.
Sometimes the objective of airfoil design can be stated more positively than, "fix the worst things". We might try to reduce the drag at high speeds while trying to keep the maximum CL greater than a certain value. This could involve slowly increasing the amount of laminar flow at low Cl's and checking to see the effect on the maximum lift. The objective may be defined numerically. We could actually minimize Cd with a constraint on Clmax. We could maximize L/D or Cl1.5/Cd or Clmax / Cd@Cldesign. The selection of the figure of merit for airfoil sections is quite important and generally cannot be done without considering the rest of the airplane. For example, if we wish to build an airplane with maximum L/D we do not build a section with maximum L/D because the section Cl for best Cl/Cd is different from the airplane CL for best CL/CD.
Another type of objective function is the target pressure distribution. It is sometimes possible to specify a desired Cp distribution and use the least squares difference between the actual and target Cp's as the objective. This is the basic idea behind a variety of methods for inverse design. As an example, thin airfoil theory can be used to solve for the shape of the camberline that produces a specified pressure difference on an airfoil in potential flow.
The second part of the design problem starts when one has somehow defined an objective for the airfoil design. This stage of the design involves changing the airfoil shape to improve the performance. This may be done in several ways:
1. By hand, using knowledge of the effects of geometry changes on Cp and Cp changes on performance.
2. By numerical optimization, using shape functions to represent the airfoil geometry and letting the computer decide on the sequence of modifications needed to improve the design.