Maximum Dynamic Lift Coefficient
The airplane people discovered some time ago that when a wing’s angle of attack is increased rapidly, it can momentarily generate a higher maximum lift coefficient than it could if the angle of attack were increased slowly. This also applies to helicopter blades. A good review of the phenomenon, which is referred to as dynamic stall, will be found in reference 6.27. Figure 6.20 shows this dynamic overshoot as measured during ramp-type angle-of-attack increases in the two- dimensional tests reported in reference 6.28. The overshoot can be related to the change in angle of attack during the time required for the airfoil to travel one chord length. For airfoils that stall first at the leading edge, the dynamic overshoot is attributed to two effects: the delay in the separation of the boundary layer, and the momentary existence of a vortex shed at the leading edge after the boundary layer does separate. These effects are discussed in reference 6.29. The delay in
FIGURE 6.18 Effect of Mach Number on Maximum Lift
Source: Lizak, “Two-Dimensional Wind Tunnel Tests of an H-34 Main Rotor Airfoil Section," USA TRECOM TR 60-53, 1960; Davenport & Front, “Airfoil Sections for Helicopter Rotors—A Reconsideration," AHS 22nd Forum, 1966; Racisz, "Effects of Independent Variations of Mach Number and Reynolds Numbers on the Maximum Lift Coefficients of Four NACA Six-Series Airfoil Sections,” NACA TN 2824, 1952.
separation corresponds to the finite time required for the aft edge of the separation bubble to move forward to its bursting position. This time delay is lengthened if the airfoil is pitching nose up, for two reasons. First, the motion raises the position of the stagnation point, as shown in Figure 6.21, and produces an effect similar to drooping the nose, which was shown to be beneficial for static stall. Second, the nose-up pitching motion causes the boundary layer to develop a fuller and more stable profile, which resists separation. A quantitative evaluation of the second effect is given in reference 6.30. If the airfoil has plunging motion instead of pitching motion, the maximum angle of attack occurs while the airfoil is descending, and thus the nose droop and pressure gradient effects are detrimental instead of beneficial. Nevertheless, an airfoil in plunge still exhibits a dynamic overshoot, though not as great as that of the airfoil in pitch.
Even after the leading edge separates, the airfoil can momentarily still generate high lift as a result of a vortex that is shed at the leading edge at the
instant of stall. The vortex travels back over the top of the airfoil at approximately half of the free stream velocity, according to reference 6.31, carrying with it a low – pressure wave that accounts for the very large lift coefficients shown in Figure 6.20.
Airfoils that stall first at the trailing edge also exhibit a dynamic overshoot, though considerably less than those airfoils that have leading edge stall. For example, reference 6.32 shows that a 16% thick airfoil, which would be expected to have trailing edge stall, has approximately half the dynamic overshoot of a 9% thick airfoil, which would be expected to have leading edge stall. The favorable effect of pitching motion for trailing edge stall is apparently the thinning of the boundary layer near the nose, which has a beneficial effect extending to the trailing edge. .
Wind tunnel tests of oscillating, two-dimensional airfoils are reported in References 6.23, 6.33, 6.34, and 6.35. The first set of tests used modified NACA 0012 and 23010 airfoils and oscillated them in sinusoidal pitch and plunge motions at Mach numbers from 0.2 to 0.6; the second set used modified NACA 0006 and 13006 airfoils oscillating only in pitch through the same Mach number range; and the third set used a NACA 0012 in sinusoidal and sawtooth pitch oscillations at a Mach number of about 0.3. The tests of reference 6.35 used several modern airfoils, but at low Mach numbers. Some of the primary effects of varying the test parameters are shown in Figure 6.22. The first set of comparisons shows that oscillations entirely below stall or entirely above stall have only small dynamic effects, but that oscillation through stall produces a hysteresis loop in which stall is
Angle of Attack, deg
FIGURE 6.20 Lift Characteristics with Ramp Changes in Angle of Attack
Source: Ham & Garelick, “Dynamic Stall Considerations in Helicopter Rotors," JAHS 13-2, 1968.
reached late on the upstroke and is induced early on the downstroke. Reference 6.36 suggests that the lower limit of stalled lift on the downstroke is approximately the static maximum coefficient of a flat plate. Note that the lift coefficient above the stall angle of attack in Figure 6.22 does not go below 0.6. The second set of comparisons shows that the dynamic overshoot is a function of the frequency of oscillation. The frequency is expressed in terms of the reduced frequency, k: where 0) is the frequency of oscillation in radians per second, c/2 is the semichord, and V is the local velocity. Physically, k is the portion of the oscillation cycle, in radians, which occurs during the time the air travels half of a chord length over the airfoil. For a blade element, the velocity is the tangential velocity, UT. In order to
FIGURE 6.21 Effect of Pitch Motion on Location of Stagnation Point
obtain an understanding of the approximate magnitude of kf assume a rotor in hover is undergoing a once-per-rev pitch change. Then the reduced frequency at the tip is one-half the amount of azimuth subtended by the chord:
k =—1— np 2 (R/c)
or, for the example helicopter with a 30-ft radius and a 2-foot chord:
The reduced frequency will be higher for inboard blade elements, for blade elements on the retreating blade in forward flight, or for blade elements that are being subjected to higher frequencies because they are coming close to a series of vortices shed by the tips of previous blades or being affected by blade torsional oscillations. The effect of increasing the Mach number is shown in the third set of comparisons in Figure 6.22. At low Mach numbers, the airfoil has leading edge stall; thus changes in conditions at the nose are significant in determining the amount of dynamic overshoot. At higher Mach numbers, however, where stall is caused by separation behind a shock wave, the stall occurs before the nose conditions become critical, and thus the pitching motion produces less overshoot.
A blade element in flight experiences plunge motion where the leading and trailing edges have vertical velocity in the same direction, as well as pitch motion where the leading and trailing edges have vertical velocities in opposite directions. The equation for the local angle of attack is:
Effect of Mean Angle of Attack, ap
M = A, к = .12
Effect of Reduced Frequency, к
M – .4, a0 = 12.5°
Effect of Mach Number, M к = .12
Source: Liiva, Davenport, Gray, & Walton, “Two-Dimensional Tests of Airfoils Oscillating Near Stall,” USAAVLABS TR 68-13, 1968.
a = 0 + tan 1 jy-
and thus the rate of change of angle of attack is:
. a _XU? a = 0 + tan —
where the first term is the rate of pitch (up) and the second is the rate of plunge (down). It is of interest to note that in the third quadrant, where the blade is pitching up, it is also plunging up as a result of the effects of coning and of the longitudinal gradient of induced velocity. Figure 6.23 shows that for the NACA 0012 airfoil, the dynamic overshoot corresponding to plunge is considerably less than that due to pitch.
Sources: Liiva et al., “Two-Dimensional Tests of Airfoils Oscillating Near Stall,” USAAVLABS TR 68-13, 1968: Gray & Liiva, “Wind Tunnel Tests of Thin Airfoils Oscillating Near Stall,” USAAVLABS TR 68-89, 1969.
Sources: Liiva, Davenport, Gray, & Walton, “Two-Dimensional Tests of Airfoils Oscillating Near Stall,” USAAVLABS TR 68-13, 1968; Gray & Liiva, “Wind Tunnel Tests of Thin Airfoils Oscillating Near Stall,” USAAVLABS TR 68-89, 1969.
A limited indication of the effect of the airfoil physical parameters on dynamic overshoot is given by Figure 6.24, which shows test values of dynamic overshoot in both pitch and plunge as a percentage of the static maximum lift coefficient for the four airfoils of references 6.23 and 6.33.
Another set of dynamic test results for a number of modern helicopter airfoil sections oscillating in pitch at a Mach number of 0.3 is reported in reference
6.35. Figure 6.25 summarizes the results. One conclusion that can be drawn is that
FIGURE 6.25 Dynamic Stall Characteristics of Several Helicopter Airfoils
Source: McCroskey et al., “Dynamic Stall on Advanced Airfoil Sections,” JAHS 26-3, 1981.
the dynamic overshoot in lift coefficient is nearly independent of the airfoil shape, varying from 0.7 to 0.8 for this group.
A number of ways have been suggested for representing the dynamic overshoot in actual practice. Several sophisticated analytical methods are given in references 6.37, 6.38, 6.27, and 6.39. These use potential flow and boundary-layer equations to predict the separation angle. Although they might be suitable for predicting the performance of an oscillating airfoil in a wind tunnel, they are much too complicated to use in a rotor analysis program. Three simpler methods based on empirical studies of oscillating airfoil wind tunnel data have been used in rotor programs and, at this writing, all must be considered valid, although none of them has undergone rigorous comparison against a wide range of rotor experimental data.
. The method of reference 6.40 as expanded in references 6.31 and 6.41 calculates the overshoot of angle of attack as a function of the pitch rate, which is taken as the rate of change of the calculated angle of attack. The method is based on a series of analogies backed up by the selected test results. It does suffer, however, from a confusing writeup and a lack of explicitly stated correlation factors.
The method of references 6.34 and 6.42 is based on the observation that the overshoot is a function not only of the angle of attack and its velocity, but of the acceleration as well. Tabulated influence factors as a function of these three parameters are stored in the computer, or curve-fitting equations are used based
on analysis of test data. Hints for extrapolation to test conditions or airfoils not tested are given in reference 6.42.
The method of reference 6.43 is also based on test data. It uses the angle of attack and its velocity in empirical equations to approximate the dynamic overshoot. This is the method used for calculations in this book and is explained in detail in Chapter 3 under "Unsteady Aerodynamics and Yawed Flow." This method relies on conclusions derived from wind tunnel results that the angle of attack corresponding to the maximum lift coefficient in a dynamic situation can be directly related to the parameter: уrd/2V. For this purpose, the rate of change of angle of attack, d, is defined as the rate the airfoil had when it reached its maximum lift. To evaluate fcdjw from wind tunnel data of an airfoil oscillating at a frequency, (0, about some average angle of attack, a0, through an amplitude, ± Да, the instantaneous angle of attack is:
а = а0 + Да sin ш and
d = Дат cos (о/
At the instant of stall:
• A • —і I ^stall a0
а = Дао) cos sin 1
From the definition of the reduced frequency, k, previously used:
The test data for four different oscillating airfoils tabulated in references 6.23 and 6.33 include all of the factors required to evaluate у/cd/2V. (Note that the only usable test points are those for which the maximum lift was reached before the maximum angle of attack, since at that point the rate was zero.) Strictly speaking, the dynamic angle of overshoot is the difference between the measured angle of attack at maximum lift for the dynamic and static test conditions, but sometimes the lift curve slope is reduced at high angles of attack so that the magnitude of the angle of attack overshoot is much greater than that
corresponding to the overshoot of maximum lift coefficient. For use in the analysis, which assumes that the increase in maximum lift coefficient is directly proportional to the overshoot of angle of attack, an effective stall angle has been defined based on the measured maximum lift coefficient; the slope of the lift curve; and, in the case of the cambered section, the angle of zero lift:
„ max. „ j
Cl stall ff = + С1ц>
In Figure 6.26, the data for the V0011 and the V23010-1.58 are plotted. The slope of the lines through the data points is the function, y, where:
ttstall – dstaU = Ай = Y staueff Staustatic 1
The values of у for pitching oscillations from Figure 6.26 have been plotted on Figure 6.27 as a function of Mach number. Both the values and the trend with Mach number are different from those indicated by reference 6.43 which analysed the same test data. The differences arise primarily from the fact that reference 6.43 plots actual stall angles at which maximum lift is achieved, whereas in Figure 6.26 the effective stall angle is plotted. Another difference is apparently in the interpretation of the test data for Mach number of 0.6. This can be illustrated by examining the dynamic stall hysteresis loop at M = 0.6 in the lower right-hand corner of Figure 6.22. The angle of attack overshoot can be referenced either to the initial static stall at about 6° or to the final static stall at about 15°. In the case of the first, there is considerable angle of attack overshoot; in the case of the latter, none at all as was apparently assumed in the studies of references 6.43 and 6.44. The method used in this book was based on the 6° stall angle as being a more realistic value in light of how the results are used in the analysis.
Confession: The forward flight charts of Chapter 3 were computed before this study was made. The gamma function used was similar to the lines defined by the solid points (and dotted lines) in figure 6.27. Fortunately, the results are about the same for the blade tips at the tip speed ratio range of 0.25 to 0.35, which includes most conventional high-speed flight.
A comparison of the overshoot during pitch oscillations and plunge motions is given in reference 6.45 for M = 0.4. The general conclusion is that at this Mach number the gamma functions are not significantly different for the two types of motion, and thus they do not have to be treated differently in the analysis.