The Optimum Airfoil for High Lift

Stratford, in References 3.23 and 3.24, examined both theoretically and experimentally the possibility of diffusing a turbulent boundary layer in such a way as to produce zero wall shear. Known as “imminent separation pressure recovery,” Stratford found that it is indeed possible, with the proper pressure gradient, to maintain a velocity profile along a diffuser such that du(y)ldy is equal to zero at the wall. н(у) is the velocity in the boundary layer parallel to the wall and is a function of the distance, y, from the wall. With the velocity gradient at the wall equal to zero, the boundary layer is just on the verge of separating, since a negative value of this gradient will result in reverse flow, as illustrated in Figure 3.41.

In the abstract to Reference 3.24, Stratford states:

“No fundamental difficulty was encountered in establishing the flow and it had, moreover, a good margin of stability. The dynamic head in the zero skin friction boundary layer was found to be linear at the wall (i. e., и °c ym), as predicted theoretically in the previous paper. (Author’s note, Stratford is referring to Ref. 3.23.)

The flow appears to achieve any specified pressure rise in the shortest possible distance and with probably the least possible dissipation of energy for a given initial boundary layer. Thus, an airfoil which could utilize it immediately after transition from laminar flow would be expected to have a very low drag."

The Stratford imminent separation pressure recovery was adopted for airfoils by Liebeck and Ormsbee (Ref. 3.25) and was extended later by Liebeck (Ref. 3.22). Using variational calculus, optimum chordwise pressure distributions for the upper and lower surfaces are prescribed that are modified slightly by additional constraints not present in the optimization process. Specifically, the optimum C„ distributions are modified in order to (1) close the airfoil contour at the trailing edge, (2) round the leading edge to allow operation over an angle-of-attack range, and (3) satisfy the Kutta cohdition at the trailing edge.

The resulting modified form of the optimum pressure distribution is compared with the optimum distribution in Figure 3.42. Beginning at the stagnation point, the flow is accelerated up to a so-called rooftop region over which the velocity, and hence the pressure, is constant. Following the rooftop region, the Stratford pressure recovery distribution is employed to reduce the velocity over the upper surface to its value at the trailing edge.

One such airfoil design is presented in Figure 3.43 (taken from Ref. 3.22). Included on the figure is the pressure distribution to which the airfoil was designed. Test data on this airfoil obtained at a Reynolds number of 3 x 106 are presented in Figure 3.44a and 3.44b. Although this configuration is referred to by the reference as a-“turbulent rooftop” case, transition does not occur until the start of the Stratford pressure recovery. In this case the performance of the airfoil is seen to be good from the standpoint of Cimax and Cd. The drag coefficient remains below the value of 0.01 over a wide range of Ci values from 0.6 to 1.6.

Artificially producing transition near the leading edge severely com­promises and Cd, as shown in Figure 3.441». Still, by comparison with the standard NACA airfoils, the Liebeck airfoil appears to offer superior per­formance at low speeds and, in the future, may find application to general aviation aircraft. One possible drawback in this regard is the sharp drop in its lift curve at stall.