Maximum Static Lift Coefficient at High Mach Numbers

The test results discussed so far are in the Reynolds number range of interest to blade designers but were obtained at essentially zero Mach number, whereas the blade elements that are likely to stall on an actual rotor operate at Mach numbers from 0.25 to 0.5. The effects of compressibility on the lift characteristics of the NACA 0012 airfoil are shown in Figure 6.12 from reference 6.10. Even at relatively low Mach numbers, the local velocity over the surface can exceed the speed of sound, giving supercritical or mixed flow as indicated by the hatched lines. For this condition, the air returns to subsonic flow before reaching the trailing edge by passing through a shock wave. If the velocity is only slightly supercritical, the shock wave will be weak and its primary effect on maximum lift will be to thicken the boundary layer and thus increase the tendency toward early trailing

edge stall. This is shown by the change in stall characteristic between 0.3 and 0.4 Mach number. For high supercritical velocities, the shock wave may be strong enough to cause the boundary layer to separate entirely, thus producing shock stall. Figure 6,13 shows measured pressure distributions for selected angles of attack at

0. 3, 0.4, and 0.65 Mach numbers. It may be seen that at the lower Mach numbers the large pressure peak generated at maximum lift is rapidly destroyed as the airfoil stalls. At higher Mach numbers the height of the pressure peak is limited by the inability of the flow to sustain local Mach numbers above about 1.4. At an angle of attack of 18° and a Mach number of 0.65, the 0012 has extreme shock stall. Despite this, its lift coefficient is high and is still rising, leading to the observation that the term maximum lift coefficient loses its significance in these circumstances. A good survey of the influence of Mach number on maximum lift is presented in reference 6.11.

FIGURE 6.12 Effect of Mach Number on Lift Characteristics of 0012 Airfoil

Source: Lizak, “Two-Dimensional Wind Tunnel Tests of an H-34 Main Rotor Airfoil Section,” USA TRECOM TR 60-53, 1960.

Most modern jet transports cruise at speeds such that a shock wave stands just ahead of the quarter-chord of the wing. The shadow of this shock wave can sometimes be seen when the wing is pointed toward or away from the sun.

Even at high Mach numbers, variations in Reynolds numbers are significant. Wind tunnel tests of several six-series airfoils at various Reynolds numbers and Mach numbers are reported in reference 6.12, and the results for one of these, the NACA 64-210, are shown in Figure 6.14. Since Mach number and Reynolds number are directly related for a given chord, lines of constant chord can be plotted across the family of curves. For sea-level, standard conditions:

.14 R. N./106

M =———————–

Reynolds FIGURE 6.14 Effect of Chord, Mach Number, and Reynolds Number, on Maximum Lift Coefficient of One Airfoil

Source: Racisz, “Effects of Independent Variations of Mach Number and Reynolds Number on the Maximum Lift Coefficients of Four NACA 6-Series Airfoil Sections,” NACA TN 2824, 1952.

It may be seen that the chord has a strong influence on the effect of Mach number on maximum lift. The larger the chord, the higher is the detrimental effect. As a matter of fact, tests with very small chord models may show beneficial effects. This is illustrated by the test data on Figure 6.15 from reference 6.13 where the effective chord varied from 5 inches at low Mach numbers to 3.8 inches at high Mach numbers. This figure also shows a rather surprising result—that for these test conditions the 64A010 airfoil has a maximum lift coefficient that is somewhat lower than the thinner 64A006 airfoil. The probable reason is that both these airfoils are experiencing pure thin airfoil stall, which is not significantly affected by thickness ratio at low Reynolds numbers. A maximum lift coefficient of 0.8 is typical of a flat plate or of a NACA 0012 tested backwards, according to references 6.14 and 6.15.

The steady reduction of maximum lift coefficient of large chord airfoils with Mach number above.3 shown on Figure 6.14 is not necessarily a trait of all airfoils. The type of airfoil known as supercritical or peaky can actually experience an increase in maximum lift coefficient with Mach number. A supercritical airfoil is one on which the nose is shaped such that the strength of the shock wave is reduced by slowing the air ahead of it through a favorable arrangement of expansion and

Effective Chord, in. 5.0 4.5 3.8 FIGURE 6.15 Maximum Lift Coefficients of Two-Dimensional Airfoil Models with Small Chords

Source: Stivers, “Effects of Subsonic Mach Number on the Forces and Pressure Distributions on Four NACA 64A-Series Airfoil Sections at Angles of Attack as High as 28°, NACA TN 3162, 1954.

compression waves, as shown in Figure 6.16. The initial expansion wave is generated by the high nose peak on the pressure distribution, which is reflected from the sonic line as a compression wave and then from the airfoil surface as another compression wave. If the shock wave is located in the region influenced by the second compression wave, its strength will be reduced as a result of the lower local velocity, and its ability to produce shock stall by separating the boundary layer will be correspondingly reduced. If the shock wave lies at some other location, the shock stall may be more severe. Thus the favorable conditions exist only for certain combinations of angle of attack and Mach number, and a penalty may apply to operating at other combinations. The first half of Figure 6.17, from reference 6.16, shows that a 12% thick airfoil with supercritical characteristics has a higher maximum lift coefficient than a 12% thick drooped-nose airfoil only for Mach numbers above about 0.43. At low Mach numbers, the peak on the supercritical pressure distribution—and the corresponding unfavorable pressure gradient behind it—encourages early bursting of the laminar separation bubble. Results for the NACA 0012 from the same series of tests are also shown for comparison. It

may be seen that both of the cambered airfoils are better than the symmetrical airfoil, because the upper surface coordinate at the quarter chord is higher and the corresponding benefit shown in Figure 6.7 applies. Similar results for 15% thick airfoils are also shown in Figure 6.17 from reference 6.17. A complete discussion of supercritical airfoils is given in reference 6.18, and procedures for designing these airfoils for rotors are described in references 6.19 and 6.20.

Figure 6.18 shows the effect of Mach number on the measured maximum lift coefficients of several airfoils with nominal 2-foot chords as reported in references 6.10,6.12, 6.21, and 6.22. Those airfoils that have good characteristics at low Mach numbers appear to retain their relative advantage in the Mach number range of the retreating blade: from 0.25 to 0.5.

Some caution should be exercised in drawing conclusions on the relative merits of the various airfoils shown in Figure 6.18, since the results appear to be affected by the wind tunnel that produced them. Figure 6.19 shows test results for the NACA 0012 airfoil obtained from several tunnels as reported in references 6.9, 6.10, 6.22, 6.23, 6.24, 6.25 and 6.26. It is possible that the difference between tunnels is associated with the thickness of the boundary layer on the side walls and

Sources: Benson, Dadone, Gormont, & Kohler, “Influence of Airfoils on Stall Flutter Boundaries of Articulated Helicopter Rotors,” JAHS 18-1, 1973; Graham, Nitzberg, & Olson, “A Systematic Investigation of Pressure Distributions at High Speeds over Five Representative NACA Low-Drag and Conventional Airfoil Sections,” NACA TR 832, 1945.

the amount of the model affected. In any case, the differences should be resolved for the good of both the helicopter aerodynamicist and the wind tunnel engineer.