Category Helicopter Performance, Stability, and Control

The sophisticated prescribed wake digital computer program described in reference 1.5 is considered to be an investigative tool for studying hover performance, but not a practical way for making quick engineering estimates. For
this purpose, three methods are suggested, one of which should satisfy the needs of the moment:

• Combined momentum and blade element method

• Hover charts

• Adjustment of existing whirl tower data

Each method has its special advantages. The combined momentum and blade element method is flexible but tedious unless a computer is available. The chart method is fast but is restricted by the assumptions that were used in preparing the charts, which may not be strictly applicable to the rotor being analyzed. The adjustment of existing whirl tower data has its greatest usefulness when the rotor being analyzed shares a special feature, such as airfoil section, with a rotor that has already been tested. Each method is described in the following paragraphs.

It is reasonable to expea that the boundary layer on the blade—being composed of molecules directly affected by the surface—would have a centrifugal force that would tend to produce a flow outboard toward the tip and that this radial flow would represent a power loss. Measurements of flow in the boundary layer of a hovering rotor reported in reference 1.17, however, indicate that the flow is slightly inboard instead of outboard. The direction of flow appears to be a function of four effects: centrifugal pumping, wake contraction, spanwise pressure gradient, and undeveloped tip vortex. In different flight conditions, one or more of these predominate to produce either inboard or outboard flow. The conclusion based on what is now known of the problem is that neglecting radial flow is an acceptable assumption.

Ground Effect

Just as with an airplane, the helicopter flying close to the ground requires less power than when it is flying far from the ground. The source of this ground effect for a hovering helicopter may be visualized by picturing an image rotor flying upside down at the same distance below the ground as the actual rotor is above it, as shown in Figure 1.38. The image wake is considered to be formed of a series of spiral vortex filaments generated at the blade tips and carried up by the image rotor induced velocity. The upward velocity induced in the plane of the actual rotor by the image set of vortex filaments can be calculated and used as a correction to the normal induced velocity term in the power equation. From a blade element standpoint, the reduction in power corresponds to the reduction of rearward tilt of the lift vector, as shown in Figure 1.39. From that figure, it may be seen that in order to maintain the same angle of attack, and thus thrust, the blade pitch must be reduced when flying in ground effect. (That the presence of the ground can influence the flow conditions at the rotor can be demonstrated at the breakfast table. The characteristics of the stream of syrup at the lip of the pitcher can be changed depending on how high it is above the pancake.)

Out of Ground Effect In Ground Effect

z/D

O. G.E.

1

.75

.5

.25

125

Symbol

о

□

A

v

0

0

Source: Knight & Hefner, “Analysis of Ground Effect on the Lifting Airscrew,” NACA TN 835, 1941.

The key to the ground-effect analysis is how much the induced velocity at the rotor disc is reduced compared to what it would be out of ground effect. Figure 1.40 shows the measured effect of the ground on one of several model rotors reported. in reference 1.18. These and similar model test results from reference 1.10 have been used to determine the induced velocity ratio as a function of rotor height using the difference in the torque/solidity coefficient at a constant thrust/solidity coefficient. The results of this analysis are shown in Figure 1.41. An extensive study of the hover performance of helicopters tested by the Army is reported in reference 1.19. The results are shown as a dashed line in Figure 1.41.

The classical studies of ground effect on hovering rotors are found in references 1.18 and 1.20. In these studies, primary emphasis was placed on the ratio of thrust in ground effect to thrust out of ground effect at constant power. For many calculations, such as for hover ceiling in ground effect, however, it is more convenient to be able to compute the power required at constant rotor thrust. The

Sources: Circles: Knight & Hefner, "Analysis of Ground Effect on the Lifting Airscrew," NACA TN 835,1941; squares: Bellinger, “Experimental Investigation of Effects of Blade Section Camber and Plan – form Taper on Rotor Performance,” USAAMRDLTR 72-4,1972; dashed line: Hayden, "The Effect of the Ground on Helicopter Hovering Power Required," AHS 32nd Forum, 1976.

out-of-ground-effect main rotor power is the sum of the profile and induced power;

If the rotor thrust is held constant while approaching the ground, the angle of attack of each blade element and the corresponding profile power can be considered to be a constant. The main rotor power in ground effect is thus:

or in nondimensional form:

This applies to an isolated rotor. An actual helicopter will generally exhibit somewhat more benefit from ground effect than that measured on an isolated rotor because of the accompanying decrease in fuselage download. Surveys made below a rotor hovering at Z/D — 0.5 and reported in reference 1.21 showed an upwash inboard of the 40% radius station. Also shown in the same reference is the download on a disc with a radius of 40% of the rotor radius and located 0.16 R below the rotor. For values of Z/D < 0.75, the disc actually produced an upload rather than a download. Further experimental evidence of the effect of the ground on download is presented in reference 1.22, where it is shown that the installation of a wing on a Boelkow BO 105 caused a significant increase in the power required to hover out-of-ground effect but no increase to hover in-ground effect. This loss of vertical drag in ground effect results in a slight modification of the equation for the change in torque:

ACJo

For the example helicopter:

Ст/^oge ~~ 0-085

and

Dv

G. W.

If it is hovering at 30% of its rotor diameter, from Figure 1.41:

0. 75

Дс8/°~-і«ь. = 0-0()0143

which corresponds to a decrease of 407 h. p. out of a total of approximately 2,000 h. p.

For studies in which the thrust ratio at constant power is required, the following equation can be used:

If the difference in profile power is considered to be negligible, this equation reduces to:

IGE

lOGE / const, thrust

The wake in ground effect is not really a steady flow, as is assumed by the theory, but a flow with large-scale fluctuations that can be felt as gusts by an observer standing near the helicopter. Reference 1.21 speculates that these fluctuations are associated with the vortex that is made up of the individual vortices from the blade roots. This vortex apparently writhes like a pinned snake along the ground, causing the entire wake to shift and wobble.

Pilots occasionally report that when hovering in ground effect, they experience random yaw disturbances. This is probably due to the effect of the root vortex as it writhes near the tail rotor. Although we cannot see the local unsteadiness in the rotor’s wake, a good analogy is the local unsteadiness in a swiftly moving river. An experimental observation of this phenomenon is reported in reference 1.23, in which an instrumented helicopter model hovering with a main rotor height of one-fourth diameter experienced random vertical stabilizer force variations of 20-30% of the mean value. The variations disappeared when the rotor height was raised to half a diameter.

Many pilots claim that the ground effect over tall grass or water is less than over a solid surface. At this time there are no test data either supporting or refuting the claim.

Stall and drag divergence are of primary interest in forward flight, where they may limit the maximum speed and the maximum maneuvering capability. There can be cases, however, when either stall or drag divergence affects hover performance. The airfoil data of the NACA 0012 airfoil of Figure 1.10 shows the envelopes of the start of stall as a function of Mach number and also the envelope of the beginning of drag divergence as a function of angle of attack. For a rotor with this airfoil section, any local combination of angle of attack or Mach number above these boundaries produces stall and/or compressibility losses. If a computing procedure is used in which the airfoil lift and drag coefficients are evaluated at each blade element, these effects will automatically be accounted for; but if a, simpler method using an average lift coefficient and a corresponding average drag coefficient is used, the effects will be neglected. In this case, however, empirical corrections can be made using the following equation which has been based on a
comparison of the measured whirl tower results reported in reference 1.1 with calculations made with simple theory.

ACg/a = O. OOlAa + 0.05 AM

where Aa is the amount the average angle of attack (in degrees) exceeds the stall boundary at a Mach number corresponding to the 75% radius station, and AM is the amount the tip Mach number exceeds the drag divergence Mach number at the average angle of attack.

The use of the NACA 0012 airfoil data in the hovering analysis is a valid assumption for normal hover conditions, since most airfoils used for rotors have

about the same lift and drag characteristics below stall and drag divergence. When an airfoil is used that has significantly different characteristics than the NACA 0012, a more sophisticated method of defining the drag coefficient than simply basing it on the average lift coefficient should be used. One method is to define the drag coefficient as a power series:

cd — cd0 + cd^ + oa2

where the coefficients are chosen to give the best fit to the experimental airfoil data. Figure 1.37 shows the NACA 0012 drag curve at 0.45 Mach number and three curves that each fit the data at 1°, 5°, and one other arbitrarily chosen angle of attack. The choice of which curve to use depends on the highest blade angle of attack for a given hover condition. Chapter 6 discusses airfoils in detail and lists data sources for a number of specific airfoils.

An airfoil that had significantly different drag characteristics than the NACA 0012 could be compared to it by assuming a rotor with ideal twist as above or its data could be used directly in the combined blade element and momentum method described later in this chapter.

The induced velocity previously derived from the combined momentum and blade element theory can also be obtained using a vortex method as shown in reference 1.13. The method is similar to that used for wing analysis, in which trailing vortices are assumed to leave the wing between adjacent wing elements and to have a strength proportional to the change in lift between the elements. In the simplest rotor vortex theory, the trailing vortices from an infinite number of blades are assumed to form concentric cylindrical vortex sheets with no wake contraction. In practice, the wake does contract producing a local distortion of the induced velocity near the blade tips. Figure 1.31 shows the location of the tip vortices from a finite number of blades with and without wake contraction. It may be seen that the contraction of the wake is such as to make the older vortices force the youngest vortex up toward the rotor plane. As a matter of fact, photographs of the

FIGURE 1.31 Tip Vortex Locations with and without Wake Contrac­tion tip vortex made on a humid day at Sikorsky and reproduced in reference 1.14 have shown that in some cases the tip vortex remains in the tip path plane until the next blade actually strikes it, as sketched in Figure 1.32. The changes in the local induced velocity due to the proximity of the vortex cause large discontinuities in the angle of attack distribution, as shown in the top portion of Figure 1.33, which is based on calculations made by a sophisticated computer program as reported in reference 1.5. The lift vectors near the tip have more rearward tilt, and in some cases the tip is stalled compared with the simpler predictions. The distortions affect the distribution of power along the blade, as shown in the bottom portion of Figure 1.33. An attempt to compensate for the high angles of attack near the tip is reported in reference 1.15. It has resulted in the design of a local region of high and nonlinear blade twist on the Sikorsky Blackhawk.

It is to be hoped that a simple and accurate analytical computing method accounting for vortex interference will eventually be developed. In the meantime it is suggested that the momentum method be used with an empirical correction factor. The suggested factor is based on a study of data presented in reference 1.5, which reports on tests of a family of model rotors that had variations in number of blades, twist, blade aspect ratio, and tip speed. The results of this study are summarized in Figure 1.34 as the ratio of the measured power to the power calculated by the momentum method—accounting for variable tip loss factor and

wake rotation—plotted against a parameter that is the product of disc loading and the thrust/solidity coefficient. This parameter has little justification in logic but gives better correlation than either disc loading or Ct/g alone. It does reflect the idea that the wake contraction effect starts as an induced phenomenon that eventually results in stall of local blade elements. Similar results are obtained using the full-scale rotor performance data of reference 1.14. The use of the suggested empirical correction in Figure 1.34 eliminates the number of blades, twist, blade aspect ratio, and tip speed as significant parameters in the vortex interference problem—an assumption that appears to be justified but will warrant continuous review in light of further experience.

The tip vortices are also responsible for another characteristic of hover performance—the relative unsteadiness of the condition. Smoke studies reported in reference 1.5 show that distinct tip vortices can be traced down into the wake only about one radius. At that point, they tend to couple together in a random way, which sometimes reinforces and sometimes cancels the vortex effect. A similar observation reported in reference 1.16 reveals that usually only four well – defined tip vortices can be identified under the reference blade, regardless of the number of blades in the rotor. Figure 1.35 shows Schlieren photographs of the wake of a model rotor in a quiet room. The single exposure shows the third vortex beginning to twist about itself, which soon resulted in self-destruction. Further observation shows that when the vortex does begin to dissipate, the wake ceases to contract and begins to expand instead, as shown in the third photograph of Figure

1.35. The multiple-exposure photograph shows that the vortex paths do not repeat themselves and that at least once during the 15 revolutions a very significant transient excursion occurred. This type of randomness manifests itself as unsteadiness in the inflow at the rotor disc and in both measured thrust and power.

CD

$

О

0.

■О

CD

N

«

Е

о

Z

Source: Clark, “Can Helicopter Rotors Be Designed for Low Noise and High Perform­ance?" AHS 30th Forum, 1974.

Figure 1.36 presents a 14-second record of the thrust and power variations of the AH-56A rotor on the Lockheed whirl tower. The variations amount to approximately ±3% for thrust and ±6% for power. This type of variation makes hover performance difficult to measure accurately and probably accounts for much of the test scatter in Figure 1.34. Yet another illustration of the nonsteadiness of hover is given by reference 1.14, where it is shown that even on a whirl tower in

Source: Data from Landgrebe, “An Analytical and Experimental Investigation of Helicopter Rotor Hover Performance and Wake Geometry Characteristics,” USAAMRDL TR 71-24, 1971; and Jenney, Olson, & Landgrebe, “A Reassessment of Rotor Hovering Performance Prediction Methods," JAHS 13-2, 1968.

winds of less than 5 knots, the vortex interaction occurs over only one-quarter to one-half of each revolution.

The wake has two effects operating on it that tend to produce rotation. One is the profile drag, which brings some air molecules up to the speed of rotation before losing them to the wake, just as a truck on the highway produces a following wake. The energy associated with this rotation is accounted for in the computation of profile power. The other cause of rotation is an induced effect that was not accounted for in the foregoing analysis. This rotation may be visualized by examining an idealized rotor wake made up of tip and root vortices which form helixes under the rotor. From Figure 1.28 it may be seen that the horizontal components of circulation in both the tip and the root vortices are oriented in such a manner as to induce wake rotation in the direction of rotor rotation. The equation for the energy associated with this induced rotation, or swirl, can be derived from momentum considerations similar to those used in the previous derivations. The figure shows a rotor wake that has no rotation above the disc but which has a rotation, 0), below it. The change in total pressure is:

Source: Bellinger, “Experimental Investigation of Effects of Blade Section Camber and Planform Taper on Rotor Performance,” USAAMRDL TR 72-4, 1972.

= lAow„„C + ІР’Ї + 1Р(®Г)2] – [P„p»»,C + iP»f]

or:

Лр = Лр»»с + 5p(t*>r)2

The change in static pressure can be related to a change in induced power through the familiar induced power expression:

APi=ATvl

or

P і, =T

‘thrust

The induced power associated with rotation is:

ДР, • rotation

Assuming a uniform induced velocity, vb for this analysis:

which can be rewritten:

An expression for (0/Cl as a function of r/R can be derived by writing Bernoulli’s equation for air flow relative to the blade just above and just below the rotor disc:

or

Astatic = !pf2[2ft(0 ~ CO2]

but

Д/W = D. L. = pCr(M)2

thus

Note that (0 is imaginary if (r/R)2 < 2CT. To avoid this in the integration, the lower limit can be set to f2CT with little loss of validity:

The integral has been evaluated as a function of CT and is plotted in Figure 1.29. Also shown are the results of two more rigorous analyses from references

1.11 and 1.12, which were made with different assumptions but which resulted in nearly the same values as the approximate method. For the example helicopter in

hover, the value of CT is 0.0073 and the corresponding induced power due to wake rotation is about 2% of that corresponding to thrust. This correction is marginally significant for this rotor but would be significant for more heavily loaded rotors or propellers used for static thrust.

The aerodynamicist will occasionally be asked to calculate the rotational velocity in the wake to define conditions at the engine inlet and exhaust or in front of rocket pods. The equation for the induced rotation, (r(0)„ is

Source: Test points are from Boatwright, “Measurements of Velocity Components in the Wake of a Full-Scale Helicopter Rotor in Hover,” USAAMRDL TR 72-33, 1972.

The corresponding equation for the rotational velocity due to profile drag can be derived from the momentum equation in the annulus of Figure 1.18:

F = (m/sec)(Ai>)

or

dAD0 = (pAr2nry1)(cor)0

so that:

Using the fact that

v i = ClR

the induced and profile terms may be combined to give an equation for the rotational velocity at the rotor disc:

The equation can also be used below the rotor disc and for rotors that do not have ideal twist if the local induced velocity is used in place of vv For example, during the whirl tower tests reported in reference 1.4, both the vertical and the rotational velocities were measured at a location.1 R below the rotor. Figure 1.30 shows these measured velocities and the rotational velocity calculated from the vertical velocity. The correlation indicates that the method is adequate, at least for the region in which engines or rocket pods would be located.

Constant chord blades are easy to design and to manufacture, but tapered blades can be made to be more efficient aerodynamically. A special combination of taper and twist can produce not only the uniform induced velocity that is the special characteristic of blades with ideal twist, but can also make the local angle of attack constant, thus giving the opportunity to operate each blade element at the airfoil’s most efficient angle of attack where c]/2/cd is a maximum.

If a constant angle of attack, and therefore a constant lift coefficient, is to be maintained along the blade, then the increment of thrust on the annulus of Figure

1.18 must be:

AT = b ^ (f2r)V, cAr

but, from the momentum theory,

or, with some algebraic manipulation,

If both v1 and ct are to be constants, then the quantity cr must be a constant. This can be accomplished by defining the local chord, c, such that:

This type of taper produces a rotor that looks like the one in Figure 1.25—one that is impractical to build but is interesting in being theoretically the most optimal hovering rotor that can be designed. If the lift coefficient is to be kept constant, then the local angle of attack must be constant where:

In order that both a and vl are constants, then the blade must be twisted so that:

This special twist distribution is shown in Figure 1.26, where it is compared with the ideal twist for a constant-chord blade. A series of calculations for the ideally twisted, constant-chord blade discussed in the previous paragraph shows that this rotor has a maximum Figure of Merit of 0.85 at an average lift coefficient of 0.94.

If the angle of attack had been a constant, the Figure of Merit would have been the value from the equation for maximum Figure of Merit:

Thus ideal taper can increase the maximum hover performance about 4% over a blade with constant chord and ideal twist. Since the ideally tapered blade is impractical to build, it is sometimes approximated with a blade that has either a linear taper or a constant chord out to some radius station and a linear taper from that point. Such a rotor, when properly twisted, can achieve a portion of the benefit of an ideally tapered and twisted blade. Even without twist, taper is aerodynamically beneficial. A study of the equation for the nonuniform induced velocity distribution will show that a constant induced velocity can be achieved with a constant pitch if the chord is inversely proportional to the blade station, as well as for a constant chord with the pitch inversely proportional to the blade station. Reference 1.10 reports a comparison of two model rotors, each with untwisted blades, which were identical except that one was untapered and one had a taper ratio of 2 to 1. As can be seen in Figure 1.27, the rotor with the tapered blades had about a 10% performance advantage at low and moderate thrust values but suffered earlier stall. Reference 1.10 attributes the earlier stall to a larger amount of tip vortex interference, but it is also possible that the lower tip Reynolds number resulted in a significantly lower maximum lift coefficient.

A rotating blade may be subjected to torsional moments that can modify the twist distribution significantly from the nonrotating, built-in twist. Reference 1.8 reports measured "dynamic twist” of up to 5° on a full-scale whirl tower rotor. Several sources of torsional moments can be identified. One is the airfoil’s aerodynamic pitching moment about the quarter chord, which is a function of blade camber and the local combination of angle of attack and Mach number. Figure 1.23 shows the measured pitching moment characteristics of a symmetrical and of a cambered airfoil from reference 1.9. The plotted parameter is the product of the pitching moment coefficient and the Mach number squared, which is proportional to the actual pitching moment. Even for the symmetrical airfoil, th^ pitching moments are not small except at combinations of low angles of attack and low Mach numbers. Another significant source of a torsional moment is th^ position of the airfoil aerodynamic center with respect to the blade flexual axis. Ary

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.

aerodynamic center position forward of the flexual axis will result in nose-up twisting moments as lift is increased. Some designers use swept-back tips to counteract this effect.

The last effect is a centrifugal flattening moment sometimes known as the tennis racket effect, named for the tendency of a tennis racket to try to align its plane with the plane of rotation as it is swung in an arc. The forces and moments due to centrifugal forces acting on a blade with positive pitch are shown in Figure 1.24. It may be seen that the forces acting on the mass elements at the leading and trailing edges produce a torsional moment that tends to twist the blade toward flat pitch. This moment not only twists the blade but also produces a control moment that must be counterbalanced to hold the blades at a positive pitch. This moment may be reduced by the use of balancing weights located perpendicular to the blade

FIGURE 1.24 Twisting Moments Due to Centrifugal Forces

chord, as shown in Figure 1.24. These are usually called Chinese weights, for reasons perhaps better left unexplained.

Dynamic twist has proved to be a problem in correlating measured thrust with measured collective pitch, but since the thrust and the power are both affected to about the same degree, small amounts of dynamic twist have little effect on the power-to-thrust relationships.

The primary effect of nonideal twist is to require more induced power than ideal twist. Figure 1.20 shows calculated distributions of pitch, induced velocity, angle of attack, and drag loading for several values of twist for the rotor of the example helicopter at its design gross weight. Several observations may be made about this series of plots:

• Ideal twist gives constant induced velocity and constant induced drag loading.

• A linear twist of —20° comes closest to simulating ideal twist.

• All the linear twist curves have approximately the same pitch at the 75% blade station. (This is a good rule of thumb for all thrust levels.)

• The profile drag loading with ideal twist is high near the root because of high angles of attack.

Figure 1.21 shows the effect of twist on hover performance of the example helicopter. From this series of plots, it may be observed that:

FACTORS AFFECTING HOVERING PERFORMANCE 41

• Increasing the twist decreases the induced torque from 8% more than ideal twist to about 2% more.

• It is possible to have too much twist, especially at the light disc loadings.

• A rotor with linear twist in general has less profile torque than a rotor with ideal twist because of the high inboard angles of attack noted on the previous figure.

• Going from no twist to ideal twist can raise the Figure of Merit about 5%.

• Most of the potential benefit of twist is realized in the first 10° of twisting.

Figure 1.22, based on references 1.5 and 1.6, shows that the theoretical effects of twist are verified by model tests.

It should be pointed out that whereas high twist is beneficial in hover, it produces high vibratory loads in high-speed forward flight and thus is usually limited to some compromise value. Currently this compromise is in the neighborhood of —5° to —16°.

One secondary twist consideration is that the negative values that are beneficial in reducing the angles of attack in powered flight are detrimental in autorotation. Reference 1.7 also presents test data showing that the optimum twist for hovering in ground effect is significantly less than is optimum for hovering out of ground effect.

Sources: Clark, “Can Helicopter Rotors Be Designed for Low Noise and High Perform, ance?” AHS 30th Forum, 1974; Landgrebe, “An Analytical and Experimental Investigation of Helicopter Rotor Hover Performance and Wake Geometry Characteristics,” USAAMRDL TR 71. 24, 1971.

The assumption of a uniform induced velocity distribution simplified the previous analysis, but in order to reflect more accurately actual conditions, it must be replaced by a nonuniform distribution. This distribution can be considered to consist of two effects, a local tip effect due to vortex interference, which will be discussed later, and an overall effect, which can be analyzed by combining the momentum and the blade element systems of analysis at an annulus of the disc, as in Figure 1.18. The increment of thrust on this annulus, AT, is:

AT = pvl2irArv2

2 3 4 5 6

Number of Blades, b

FIGURE 1.17 Effect of Number of Blades on Ideal Figure of Merit as Calculated by Two Methods

Source: Harris & McVeigh, “Uniform Downwash with Rotors Having a Finite Number of Blades,” JAHS 21-1, 1975.

where 2rrrAr is the area of the annulus and vx and v2 are the induced velocities at the rotor disc and in the remote wake, respectively. Just as in the original derivation of the momentum equation, it may be shown that: so that the equation for AT becomes:

AT = 4prr^rAr

From the blade element theory, the increment of thrust can also be written:

Equating the two expressions for AT and arranging the result gives:

a a2

4tv] + — bacv,———- rbaQc = 0

2 2

or

— — acb + J ^—acbj + 8 тЬП2га$с

8 її

This is a perfectly general equation for the induced velocity at any radius, r. It can be used with any twist distribution by using the correct value of blade pitch, 0, at the blade station, and it can be used with any blade taper scheme by using the correct value of the chord, c.

For a constant chord blade, the equation can be manipulated to give:

Note: If the analysis is being done for a rotor with cambered airfoils, 0 should be replaced by (0 — aoi) where aoi is the angle of attack for zero lift.

For this case, it may be seen that the induced velocity depends on the radius station only as the parameter, 0(r/R), varies with the radius. If the rotor has ideal twist, then this parameter is a constant being equal to the pitch at the tip:

For blades with constant chord and ideal twist, the induced velocity is a constant across the disc as was originally assumed in the momentum theory. Thus one definition of ideal twist is the twist required for constant chord blades to produce a uniform induced velocity.

The lift distribution corresponding to ideal twist and uniform induced velocity is triangular. This is in contrast to the ideal wing, which produces uniform induced velocity with an elliptical lift distribution. The equation for lift distribution written in terms of the circulation, Г, is:

= cTV = cTOr

For a triangular lift distribution on an ideal blade, the circulation is a constant; and only two trailing vortices are generated: one at the root and one at the tip.

The nonuniform distribution of induced velocity of a rotor that does not have ideal twist manifests itself in the remote wake as a nonuniform dynamic pressure distribution, which is significant in making estimates of fuselage downloads or ground erosion. Figure 1.19 shows the measured distribution from reference 1.4 for several locations downstream of a full-scale rotor with a linear twist of —4°. It may be seen that the wake contracts very rapidly, accomplishing most of the contraction within the first 10% of radius. The tests were conducted in winds of less than 3 knots, but the test results show that even winds this low can deflect the wake significantly.