Category FLUID-DYNAMIC LIFT

Because the propulsion system must react on the air to produce thrust, the velocity in the slipstream is higher than in the free stream the magnitude is directly propor­tional to the disk loading. With a high disk loading the area of the slipstream is smaller than with light loadings for a given level of thrust. Thus, thrust systems such as propellers will have a large slipstream area with a relatively small increase in velocity. For this reason the propeller slipstream of tractor type airplanes will impinge on large portions of the airplane, the area of the wing and fuselage effected depending on the configuration. Airplanes have been designed to minimize slipstream effects by using pusher propellers. A notable example is the six engine pusher propeller B-36 airplane. Many other pusher pro­peller installations have been built but usually they never reached production. In the case of the B-36 the pusher configuration was selected as it was desired to eliminate the propeller slipstream effects on the wing and to main­tain laminar flow.

With tractor installation it is possible to have an airplane configuration using ducted fans, turbofan engine or jet engines where the slipstream does not impinge on the airplane structure. Because the effects of the slipstream can be desirable as well as harmful, it may not be an advantage to eliminate all slipstream effects on the air­plane. This, of course, will depend on the particular air­plane and its performance goals.

(6) Slipstream Effects – Propellers:

a) Theodorsen, Static Propellers & Helicopter Rotors, AHS

25th Forum, 1969.

b) Jameson, A., Propeller-Wing Flow Interaction, NASA

SP-228; also NASA CR-1632.

c) Smelt & Davis, Lift Due to Slipstream, ARC RM 1788

(1937).

d) Stuper, Evaluation, Lufo 1935 p 267; 1938 p 181.

0 Bradfield, Airscrew and Wing, ARC RM 1212 (1929).

g) Kuhn, Wing-Propeller Combination, NACA T Rpt 1263

(1956); also Continuation, NASA D-17 (1959).

h) Kuhn, Propeller & Wing & Flap, NASA Memo 1-16-1959,

L.

i) AVA Gottingen, Wing in Slipstream, Erg I (1920) p 112.

6) At several propeller diameters aft of the disk, the wake expands with a 15 degree included angle like that of most streams.

7) The actual velocity imparted to the slipstream by the propeller consists of axial, radial and tangentional com­ponents of velocity, which are periodic in nature. Immedi­ately downstream of each blade the velocity increment is a maximum. In between the blades the velocity increment is a minimum, but is still more than 50% of the peak. Except at zero and low speeds the radial velocity is small and can be neglected. The tangentional component of velocity in the slipstream is in the same direction as the

W propeller rotation.

8) The increase of the axial velocity is not uniform across the disk but varies as the thrust and torque loading, as is illustrated in figure 14.

9) When the propeller is operating at an angle of attack the down going blade produces a higher level of thrust than that of the up going blade. This results in an increase of the slipstream velocity aft of the blade with the higher level of thrust.

10) Due to mixing and damping the flow irregularity found immediately aft of the propeller is expected to become more or less steady.

Slipstream Characteristics – Propellers. The flow charac­teristics in the propeller slipstream are complex because there is no duct to control the flow. The general charac­teristics are as follows:

1) The axial free stream velocity is increased; one half the increase taking place in the propeller disk, the other one half in the final wake.

2) The diameter of the slipstream is approximately equal to that of the propeller, especially at the normal forward flight conditions.

3) At zero velocity and low forward speeds the slipstream contracts between.816 and.92 depending on the loading and the forward velocity, figure 13, (6,a).

4) The slipstream contraction takes place very close to the propeller disk as illustrated on figure 13.

5) Because of the rapid contraction of the propeller slip­stream the average axial velocity in the final wake is used for determining wake effects using equation

v = T/m = T/Sp p Vі

where Sp is the propeller disk area and Vі is the axial velocity through the disk.

11) As a result of the lift produced by a propeller when operating at an angle of attack the slipstream is deflected downward, thus reducing the horizontal tail effectiveness.

Propeller Slipstream – Thrust Effects. Although the ac­tual characteristics of the propeller slipstream are complex it is generally permissible to assume a uniform increase in the dynamic pressure for finding its effects on the air­plane. Thus the dynamic pressure in the slipstream q” is

q” = q + T/S p (10)

where Sp is the area of the propeller disk = D /4.

For evaluating the propeller wing combination non – dimensional thrust coefficients based on the stream dy­namic pressure are preferred. These coefficients are based on conditions in the free stream and in the slipstream. Thus

Tc = T/q S p (11)

Tc = T/q”Sp (12)

Where Tc is the propeller thrust coefficient based on the free stream dynamic pressure and the primes indicate the coefficient based on. the dynamic pressure q” in the pro­peller slipstream. In the test and anlysis of propeller V/STOL aircraft the thrust coefficient Tc* based on the total forces of all propellers, wing area and qis used. Thus

Tc = T’/qs = ^ (13)

where К is the number of propellers and T’ = total thrust force.

Lift Due to Slipstream. When wing is immersed in the propeller slipstream an increase lift will be obtained due to the increased q. If the angle of attack is well below that for stall and the effect of the rotational velocity change is small, the lift coefficient can be described in terms of the thrust coefficient and section data by the equation

CL =(4 /о-T") (14)

Thus, the actual lift coefficient based on free stream conditions is increased over the section data.

For the analysis of V/STOL airplanes it is more con­venient to use the total wing area and number of pro­pellers in developing the overall performance charac­teristics.

CL = C " ‘(1 + Tc S/K Sp) (15)

where К is the number of propellers.

Propeller Wing Lift. The combined lift of a tractor pro­peller wing combination is higher than the sum of each component alone. The wing lift is increased by the incre­ment of slipstream velocity of the propeller, equation 10, while the lift component of the propeller normal force is increased due to the increase of upwash angle of the wing. In addition to increasing the dynamic pressure at the wing the propeller alters the angle of attack and decreases the lift slope (6,b). An example of the combined lift of the propeller wing combination is given on figure 15 as a function of К Tc. Here, К is the number of the propellers on the wing and Tc is equal to T/q Sp. Note the slope of the lift curve increases due to the direct lift effect and the velocity increase of the slipstream. The maximum lift of the combination depends only on К Tc .

On figure 15 the normal force coefficient and its slope are given for the propeller alone without the nacelle used with the wing. Note the relatively straight line variation of CNp with angle of attack for this case.

Propeller Jet Effect on Wing Lift. When the wing is operating in the propeller slipstream the effective angle of attack is changed along with the lift curve slope. Thus, to calculate the combined lift of the propeller wing combina­tion it is necessary to determine the effect of the propeller jet on the wing, in addition to the propeller normal force and the increase dynamic pressure (6,b).

If only one propeller is operating in front of a wing panel the effective jet aspect ratio is one. However, if two or more propellers are used, the jet aspect ratio is found from the equation 16 assuming the spacing is close enough so that the jet is continuous

Aj = B/H (16)

where В and H are the height and width of the jet.

From the detailed calculations of (6,b) the slope of the lift curve can be found for the wing operating in the wake of the propellers from equation 17 and 18

CWo=CL«7A + 2>/[A + Aj + (2-5/1+A)] (17)

^La/uT

Clrtl/1 + [(С, Ш/С )-1][(1-М)/(1+А^ )

Lo’1 ‘ (18)

where M = V/V,’ the ratio of the free stream velocity to the jet velocity or the velocity in the slipstream aft of the propeller. The subscript 1 represents the case where the

velocity over the wing is the jet velocity and о is the free stream velocity. Thus, equation 18 gives the lift curve slope for the free stream velocity case in terms of the slope when the free stream velocity equals the jet velocity. To find the lift curve slope at д, equation 18 must be used. Note all the values of the slope are based on q in the jet. Based on this the slope decreases as the external flow is reduced.

The drag is also effected by the jet flow on the wing. If r = C0ICLZ then

Го =r, ,76(Aj +eAj)+.53 (19) ru = r, [(r0 /r, ) + [1 + Aj -(r0/r,)]

The above equations are valid for the range of A > ViA 16. For the wing operating at zero forward speed the effectiveness is increased by an increase of the jet aspect ratio and a decrease in wing aspect ratio. Thus the lift of the wing will increase as the jet becomes narrower. This shows the advantage of several small propellers washing the wing rather than one large prop.

Total Wing Propeller Lift. The total wing propeller lift is thus calculated by finding the propeller normal force operating in the upwash produced by the wing, adding the change in the wing lift due to the jet effect of the propeller and its associated slipstream velocity. Good cor­relation with experimental data have been obtained using this procedure for unflapped wings with the propeller thrust axis nearly parallel to the wing chord line. When flaps are used the accuracy correlation is reduced but a good first approximation is obtained.

The lift force by the propulsion system may range from zero to over one hundred percent of the weight of the aircraft as in the case of helicopters. The propulsion system of STOL airplanes may generate lift directly or may induce increased levels of lift from the increased velocity of the slipstream passing over the wings and fuselage. The addition of lift with boundary layer control either by blowing or sucking is a form of propulsion lift, as power must be directly expended. In this context all types of lift could be thought of as powered lift; however, we shall define propulsion lift as that generated by the device normally used only for overcoming the drag of the airplane. The effects of the wash or slipstream on the wing lift produced by the propulsion devices will also be in­cluded in this chapter. The lift increment from boundary layer control through the use of power by internal blow­ing or sucking is discussed in Chapters V and IV.

Because the lift force produced by the propulsion device can be large it has an important effect on the stability of the airplane. This will be considered in terms of the lift, moments and interaction developed by the propulsion device.

THRUST DEVICE

Helicopter Rotor VTOL Propeller STOL Propeller Conventional Propeller Ducted Fan Lift Engine Turbo Fan Jet Engine

The thrust or the lift to power ratio of the engine types given in the above table operating at the static condition is given on figure 2. The large thrust to power ratio shown at low disk loadings is the reason for the use of these loadings on helicopter rotors. The propulsion devices given in the above table are also listed in the order of the mass flow handled. The largest devices handle the greatest mass flow and give the fluid the smallest increase in velocity since the mass flow,

m = P A A V

T – m д V

where V is the velocity through the disk area A and V is the increment of velocity imparted to the air.

Consider a propeller operating at an angle of attack as shown in figure 4. Assume the propeller has four blades and that two of the blades are at the horizontal position. From figure 4 it will be noted that when the blades are at the horizontal position an increase in lift on the blade advancing down and a decrease in lift on the blade coming up are obtained. This is the result of the increase and decrease of section angle of attack and speed on the blades going down and up, respectively. It will be seen that these forces resolve themselves into a normal force in the axis of rotation and that the forces also result in a turning or yawing moment about the shaft.

The equations for calculating the normal force and mo­ment have been derived from blade element theory in a manner such as given in (1) as follows:

For a right-hand rotating propeller with positive lift nor­mal force, the moment results in yaw couple to the left; the opposite being true for a left-hand propeller.

It can be shown for small values of A equations 2 and 3 can be reduced to

Comparisons of the test (2,c) and calculated values of normal force based on equations 3 and 4 show generally good agreement up to shaft angles of 45°, figure 5. At shaft angles above 60° the correlation is poor and more refined methods based on helicopter theories must be used.

Fn =

Figure 3. Comparison of normal force produced by wings and propellers.

К BAqAFD sin 0.7 (ao7 + 2C/0 7 cot/0 7 )/K M = (4) К’ BAqAFD2 cosA,7 (ao7 + 2Ci0 7 C0t^7 )/K, (5) where K’ = 1.1 AF = activity factor D = propeller diameter

SECTION A-A @ 8=0, 90, 180, 270 & 360°

The calculation of the propeller normal force above is based on the assumption that the slope of the lift curve slope for two dimensional flow applies and is not affected by the cyclic variation of the lift force AL, figure 4. This is a reasonable assumption up to a shaft angle of 45° as the variation of lift causes a change in the strength of the shed vortices which cancels at the blade, allowing the use of equation 3 for calculating the in plane normal force.

Figure 4. Force and velocity diagram of a propeller at angle of Attack A.

(2) Normal Force Test Data – isolated propellers: a) McLemore, Angle of Attack to 180 , NACA TN 3228 (1954). b) v. Doepp, Junkers Results, ZWB Tech Berichte 1941 p 61. c) Yaggy, 3 Propellers to or = 85°, NASA TN D-318 (1960). d) Weil, Two Propellers, NACA T Rpt 941 (1949).

(3) Upwash ahead of straight wings:

a) Roberts, Twin-Engine Airplane, NACA TN 2192 (1950).

b) Yaggy, Survey (TN 2192) and Prediction (TN 25 28), NACA.

c) Rogallo, Analysis, NACA T Rpt 1295 (1956); also for swept wings in TN 2795, 2894, 2957 (1952/53).

(4) Normal Force V/STOL Aircraft:

a) Borst, The High-Speed VTOL X-100 and M200 Aircraft, Aerospace Engineering, August 1962.

b) Borst, The X-19 VTOL Aircraft, NYAS, Vol 107, 1963.

c) Kuchemann and Weber, Aerodynamics of Propulsion, McGraw-Hill Book Co., New York 1953.

Induced Flow – Upwash. As shown in equations 3 and 4 as well as in figure 5, the flow angle into the disk of the propeller directly influences the normal force. Since pro­pellers are usually installed in front of wing where an upwash is encountered, the inflow angle into the disk is changed from free stream thus influencing the normal force. These changes are large and may be estimated based on the test data of (3) where detailed measurements of the inflow velocity have been made. Computer programs based on theory are also available for estimating the inflow angle at the disk, taking into account the effects of the wing upwash and the variation of the flow about the wing and nacelle. From these methods the inflow angle A should be determined and used in equations 3 and 4 for calculating the propeller forces and moments.

(5) Ducted Fan Lift:

a) Mort, K., 7-Foot Diameter Duct Prop Test, NASA TN D-4142.

b) Goodson, Tilting-Shrouded-Propeller VTOL Model, NASA TN D-987.

c) V/STOLConf. 1966.

d) Grunwald, Shrouded-Propeller Configurations, NASA D-995.

e) Koenig, Wingless VTOL Aircraft, TN D-1335.

The wing upwash as illustrated on figure 6 generally results in a high inflow angle at the propeller at the climb condition when the aircraft is operating at low speeds and high coefficients. This increases the normal lift force of the propeller.

Propeller Lift and Thrust. The thrust and normal force measured normal to, and in the plane of the disk can be resolved into components in the direction of flight and in the lift direction as shown on figure 3. Resolved in this manner, the lift becomes a straight line function up to very high angles of attack, figure 7. The coefficients and Cx are based on the free stream dynamic pressure and the disk area S^>, thus

ClP = (TsinA + NcosA)q Sp (6)

Cxp = (TcosA – NsinA) q Sp. (7)

The lift and axial force coefficients CLp and CXp are a function of advance ratio as noted in figure 7 and depend on the detailed characteristics of the propeller as in dicated in equations 4 and 5. Note the straight line characteristic of the lift curve. Unlike a wing there is no sudden stall such as shown in Chapter IV. This characteristic is desira­ble for V/STOL aircraft that must operate through a large variation of angle of attack.

Propeller and Rotor Moments. In addition to the thrust and normal force, a rotor or propeller operating at an angle of attack will produce yawing and pitching moments that become large, especially at high angles of attack. The yawing moment developed may be determined based on equations 3 or 5 or from wind tunnel test data, such as (2,c). The pitching moment becomes especially important at high shaft angle for V/STOL aircraft as it becomes a major factor in the design of the longitudinal control power.

For conventional airplanes the yawing moment produced by the propellers is destabilizing for tractor installations if all propellers are rotating in the same direction. This destabilizing moment is usually referred to as the fin effect (l, c) and is a function of the diameter and blade area as noted in equation 5. Since the yawing moment of a propeller at an angle of attack is dependent on the

direction of rotation it can be eliminated on a multi­propeller aircraft with the use of opposite rotation.

Normal Force V/STOL Aircraft. A propeller is a good device for developing high lift power ratios at hover, figure 2, and can produce thrust as well as lift at forward velocity. For this reason it can be used on V/STOL aircraft performing a dual function of providing thrust and lift at all flight conditions. To do this it is necessary to use opposite rotating propellers operating in pairs so the large yawing moments associated with the develop­ment of large values of normal force can be cancelled.

With such a system a linear lift force is obtained with angle of attack and it is possible to obtain lift forces of sufficient magnitutde to fly an airplane without wings (4). This principle was demonstrated with a full scale airplane, figure 8, where the propellers provided approximately 70% of the lift force at a forward speed of 175 MPH.

In nature, the dragon fly is another example of the use of propulsive lift to fly without fixed wings, figures 9 from

(4) illustrates this principle. Although nature does not provide for complete rotation of the wings the rapid oscillation of the wings does simulate the action of a dual rotation propeller, thus providing thrust and lift from the same unit.

As noted in equation 2, the propulsive lift of a rigid propeller is dependent on the shaft angle of attack, the propeller diameter, the total solidity, as well as the operat­ing condition. If the blade angle is varied cyclically the lift force can be further increased. This must be done by increasing the blade angle of the down going blade while decreasing the blade angle of the up going blade. The same mechanism used for increasing the propulsion lift could be used for control at hover and low speeds, as in the case of a helicopter.

The use of the propulsion system to generate lift requires that the shaft angle be low to obtain high values of L/D. From figure 2 it will be noted that

L/D = 1/tan A (8)

Thus, to obtain high values of lift drag the shaft angle, A, must be low. This requires that the q as shown in equation 4 be high so that the required lift is obtained at the low A angles.

Ducted Fan-Propellers. The ducted fan or propeller is ef­fective for the production of lift as well as thrust when operating at an angle of attack above zero. The ducted fan has been therefore used and proposed for a number of V/STOL aircraft including a VTOL airplane without wings (5,e). Also because of its high thrust to power and lift ratio, the ducted fan has been applied for the power lift, control and thrust of airships. In the application of ducted fans for producing lift and thrust the duet can be operated at an angle of attack or the exit flow can be deflected with vanes to obtain lift.

The ducted fan is an efficient device especially at low speeds as the interaction between the fan and the duct is mutually advantageous. The induction of flow through the duct by the fan produces a pressure reduction on the leading edge of the duct that gives a duct thrust force which adds to that thrust produced by the fan. The duct controls the flow at the fan and eliminates the radial flow and the tip losses that are associated with conventional open propellers. The elimination of the radial flow at the disk of the ducted fan is only accomplished if the duct is designed to produce the thrust level equal to that needed to obtain the ratio given on figure 9,a. This rotor thrust to the total thrust ratio, T^/Tp, is dependent on the power level of the rotor and the advance ratio, J. When the duct develops the required thrust ratio the duct flow controls the conditions at the disk and so prevents radial flow, assuming the rotor duct clearance is zero. When design­ing the duct to achieve the required thrust level a pro­cedure must be used that accounts for the thrust produced

by the rotor and its advance ratio. This procedure is also used for determining the velocity at the rotor as induced by the duct so that the performance of the rotor can be calculated using a procedure such as is given in Chapter II.

The thrust developed by the ducted fan is dependent on the blade to duct tip clearance. These losses are large unless steps are taken to seal the tip or reduce the clearance to a minimum. The efficiency of the fan cor­rected for the tip loss can be calculated from the equation

/^ =(1.00002 – 2.654Rc/R-158.35 (Rc/R)2+ 163(Rc/R|)

(9)

where /$. = the corrected duct fan efficiency /? = efficiency with zero tip clearance

Rc = clearance between the rotor tip and duct wall

R = rotor radius

Figure 9a. Rotor thrust to total thrust ratio for ducted fan required to control radial flow.

The ducted fan can be considered to be a ring wing with the fan inducing air through it, figure 10. As such the ducted propeller produces lift when operating at an angle of attack. The lift produced, (5,a) is a function of the ad­vance ratio, the thrust coefficient and the angle of attack as shown on figure 10. The lift, drag, thrust and moment coefficients given on figure 10 are based on q and the pro­jected area of the duct S. Thus

Tc = To ІЧ S ; CL = L/q s

CD = D/qS ; Cm = m/qS (9)

where S = cde, de is the exit diameter, and c the duct length.

The values of CL and Cp, Tc and moment coefficient C/n are high on figure 10 as the tests were conducted at low values of q and advance ratio. At the conditions tested the force in the drag direction is negative and so is a thrust force.

Lip Stall Ducted Fans. A ducted fan will tend to stall with an increase of the shaft angle of attack. Like a wing, stall takes place in the form of separation of the leadin g lip of the duct as illustrated in figure 11. The conditions where separation are encountered are a function of the shaft angle of attack, Reynolds number, inlet radius and the thrust coefficient Тз. Not only does the stall cause a loss of lift but it also causes a large increase in the noise level of the system and a decrease of thrust and efficiency. Because of the separation of the inner lip the propeller will encounter unsymmetrical flow which reduces the thrust and increases the blade stresses.

cx°

The angle of attack where separation can be expected on the upstream lip as a function of the thrust coefficient is illustrated on figure 11, based on tests of 4 and 7 foot diameter ducted fans and models. It should be noted that model tests of ducted fans indicate a much lower angle for stall even with large modifications of the lip radius (5,b). This illustrates the importance of q, Reynolds number and shows that model test results should be used with caution for predicting results especially where separation is in­volved.

Tests (5,c) also have shown that separation will occur on the outside of the duct as illustrated in figure 12. This separation is also a function of the lip radius, Reynolds number and thrust coefficient. Upper lip separation is not as serious a problem as the separation on the leading lip, although an increase in drag is encountered. When separa­tion takes place on the upper lip only a minor reduction in the slope of the lift curve is encountered.

Separation leading to duct stall can also be observed from plots of J/Tc a function of angle of attack as illustrated on figure 12 from (5,d). This curve shows that the separa­tion considerably reduces the thrust produced by the duct.

Duct Moments. Ducted propellers develop a pitching mo­ment which increases with angle of attack as shown on figure 10. This pitching moment appears to be mainly a function of the lift coefficient and is mainly independent of the thrust coefficient and the advance ratio.

Unlike a propeller the yawing moment appears to be low for ducted fans with increases in angle of attack at least to an angle of 30 degrees, (5,a). It would appear that the length of duct in this case is sufficient to produce a symmetrical axial flow through the duct. Thus, it would be expected that the propeller or fan in the duct has very little effect on the overall lift as the normal force is essentially zero. When the flow separates on the forward lip this may no longer be the case.

Lift Engines. Light weight engines have been developed for direct lift applications. The lift produced by the en­gines is dependent on their placement on the aircraft. As shown in (5,c) the lift can be increased up to three times that of the static thrust of the engine depending on the configuration.

Engine Lift Turbofans, Turbojets. The lift produced by isolated turbofan or turbojet engines would be smaller per unit thrust developed than either propeller or ducted fans. This would be expected as the lift is developed based on the projected area of the engine nacelle, which would be smaller per unit power than the ducted propeller. The lift produced by turbofan engines is small in comparison to that of the complete airplane and is usually only impor­tant for interference and stability effects.

Because of the higher thrust coefficients of turbofan and turbojet engines the lip stall problem is delayed to higher angles of attack than shown for ducted fans. If operation is encountered at the higher angles the intake should be treated in much the same manner as the ducted fans discussed in the last section.

Considering performance (drag), tail surfaces are basically an operational necessity, usually not contributing to lift. Also, to support them in their conventional place, the fuselage must be made longer than it would have to be, just to house payload and equipment. During some 40 years of aviation history, therefore, the idea has been advanced, discussed (27), investigated and partly con­verted into hardware, of flying without a tail in a wing – only airplane.

Stability of Airfoil (26). The pitching moment of an airfoil section, about an assumed CG (on chord line) is

C, n,-C„0 + (Ax/c)CL (48)

where x = g – a = distance between CG and AC (which may be at ^ 24% c). The stability derivative

dCm3/dCL = Д. x/c (49)

is negative = stable, as long as the CG is located ahead of the AC, ie, forward of ~ 24% of the chord. In order to obtain balance, it is now necessary to keep

Cm0+ (Дх/с) CL = zero; or CrnD= — (Дх/с) CL (50)

This can be done by changing camber such as deflecting a trailing edge flap. For positive lift coefficients, an airfoil section with reflexed camber line promises to be most efficient. Such a section (with Cmo= 0) can then be trimmed to a desired positive lift coefficient by pulling the flap up. Theoretically, stable and balanced flight can thus be obtained in a simple wing. For example, for a CG location at 22% c, the static stability derivative may be 0.22 — 0.24 = 0.02. Using derivatives as found in the Chapter IX, the required deflection of a 20% flap, would be 6 ^ – 4 CL. For each additional % of the chord through which the CG is moved forward, another A(d$/dCL ) = – 2 will be required.

Wing Sweep. A tailless airplane, simply relying on the qualities of a straight wing, is likely to be very sensitive. The wing would provide too little damping (about the lateral axis). A swept wing (having a longer length, be­tween apex and rear end of the lateral edges) would, therefore, be more suitable, using the wing tips for stabili­zation and control. Considering the areas near the tips to be a pair of horizontal tail surfaces, we then have a configuration which is basically similar to the conven­tional wing plus tail arrangement. The wing tips can and have also been used to carry a pair of vertical fins, to provide directional stability and control. One of the first such airplanes built and flown was by Dunne in 1910. This airplane was propelled by a pusher propeller, con­veniently located “in” the center of the configuration. An early British development was the “Pterodactyl” (28,a, b). Several similar airplanes (most of them gliders) were de­signed and built by Lippisch (28,c).

Wing Flaps in or near the center of the wing, could “never” be balanced in a straight-wing tailless airplane. However, as explained in the “wing” chapter, the center of the lift produced by trailing-edge flaps, is (at least theoretically) at or slightly ahead of 50% of the chord. Therefore, in-a sufficiently swept wing, inboard flaps can produce lift at a location ahead of the wing’s aerodynamic axis. As proposed in (28,b) such flaps could then also be used for pitch control; and their d(ACL)/dCL would be positive rather than negative (as explained above for ele – vons), when using them as “elevators”.

Swept-Forward Another solution to control a tailless airplane, is to sweep the wing forward, as for example, in the design in figure 39. The configuration can be trimmed by pulling up the inboard flaps. Using the average tested value dCrr/dcf0 = 0.005, a flap deflection S° = 8 is needed, for example, to balance the moment at CL = 1.0. It seems however, that conditions (and shortcomings) basically remain as they are in a swept-back wing configu­ration.

(26) Wurster, Analysis and Experimental Investigation of Stable (swept and twisted) Wings, Ybk D Lufo 1937 p 1-115. [107]

(28) Consideration of tailless airplanes:

a) Hill, Tailless Airplane, J RAS 1926 p 519.

b) Jones, Stability & Control, NACA TN 837 (1941).

c) Lippisch, “Storch” Glider, L’Aerophile Feb 1930 p 35 (NACA TM 564); also “Dreieck I”, NACA AC-159 (1932).

d) Donlan, Stability & Control, NACA T Rpt 796 (1944).

(29) Characteristics of tailless airplanes, tested:

a) Weick, Swept Wing, NACA TN 463 (1933).

b) Lippisch, “Storch” Type, Ybk D Lufo 1937 p 1-300.

c) Gates, “Pterodactyl” Design, ARC RM 1423 (1932).

d) Stone, Cornelius Glider, NACA W Rpt L-738 (1946).

e) Brewer, Northrup in F’Scale Tunnel, W Rpt L-628 (1944).

f) Tailless Airplane, see NACA W Rpts L-42 and L-50.

Delta Wings. Aircraft using a delta wing lends itself naturally to be without horizontal tail. Tunnel results on a model similar to the original Convair F-102, are shown in figure 40. Below the stall (at an untrimmed lift coefficient CL ^ 1.3) the neutral point is at ^35% “mac”. To trim the airplane, an elevon deflection d6/dCL = – (dCm/dCL)/(dCm/d6) = – 0.04/0.005 = – 8° is necessary for the CG location as during the tests. In other words, to trim the airplane to CL = 1.0, an elevon angle 6 = -8 is required. At the same time, the airplane loses some lift corresponding to ACl /Cl = (dCL/d6) (d&/dCL) = – 0.013 (8) ^ – 10% The maximum lift coefficient is reduced, corresponding to an elevon deflection 6 = – 30 required to trim.

Figure 40. Delta-wing configuration similar to Convair F-102, tested at M = 0.7 (30,a).

(30) High-speed tail less delta airplanes: a) Hewes, 60 Delta Configuration, NACA RM L54G22a. b) White, Flight-Tested, NASA Memo 4-15-59A, Douglas F4D-1. Other characteristics of same airplane, see RM A52E23/J30/J 31.

(31) “Tailless” rocket-powered interceptor airplane Me-163, see Chapter XIV cf “Fluid-Dynamic Drag”.

Figure 39. Characteristics of a tailless, swept-forward fuel-glicler design. Tested at Rc = 2 (10)5.

Interceptor. The research efforts reported in (28,c) and (29,b) eventually led to the first rocket-powered and “tailless”, operational airplane (31). Here as in many other configurations designed without a special horizontal control surface, one can argue whether a vertical surface is a tail or not. Apart from delta designs, one of the very few operational tailless airplanes is or was the U. S. Navy’s F7U-1 “Cutlass” fighter (33). To provide directional stability, this airplane carries a pair of fins (above the wing), each approximately at Уі span. Principles of tailless airplanes and some results of model tests are presented in

(32) .

Conclusion. After reviewing various aspects of tailless air­planes, it may seem that the original idea of saving wetted area (by combining, at least the horizontal tail with the wing) will not necessarily reduce drag. A larger wing area will most likely result, and larger vertical fins. It has also been explained in (27) that size and speed of the airplane are to be considered together with the problem of the tail. In particular, modern high-speed airplanes have fuselages which are large in comparison to the wings. As a conse­quence:

(a) The original idea of stowing away everything within the wing, can no longer be realized.

(b) The fuselage is available anyhow as a “boom” to carry both the vertical and the horizontal tail surfaces.

However, there can always be reasons apart from per­formance, to make a tailless airplane desirable. One such type of configuration is delta-wing fighter airplanes. See Chapter XVIII.

Trim Required. Results of flight tests on a full-scale carrier-based jet airplane are reported in figure 41. The discontinuities in the elevon deflection are confirmed by corresponding jumps in the stick force. They seem to be a consequence of the automatic leading-edge slats as shown in the illustration. For the CG location as tested, the elevon angle corresponds to d6/dCL = – 14 . Assuming that other derivatives might be similar in magnitude to those of the model in figure 40, the static stability may be

dCm/dCL = – (dCm/d6) (d6/dCL ) = – 0.005 (14) = – 0.07

in the condition as tested. The neutral point of the airplane would then be at 24.4 + 7.0 = 31.4% of the “mac”. The trimmed maximum lift coefficient (C^ := 0.8) is obtained with the help of 6 = – 11 or – 12°. It seems that the slats make the elevons more effective at higher angles of attack, although they do not help to increase the maximum operationally obtainable lift.

The basic equations of longitudinal balance and stability suggest that “any” type of combination of two “wings” may be used in designing an airplane. One unconventional arrangement is the canard type configuration, flying tail- first, so-to-speak.

Canard Principles. Many pages of this and of the next chapter, are devoted to the description of all the inter­action and interference effects to which a horizontal tail surface are subjected. From time to time, the possibility has, therefore, been investigated of placing the “stabi­lizer” ahead of the wing, where “no” disturbance may have to be expected. Another argument in favor of this arrangement is the fact that the canard surface will reach its maximum, lift at a lesser airplane angle of attack (and/or wing lift-coefficient) than a tail surface (located in the wing’s downwash). It has been stated accordingly, that the wing of the canard-type airplane would be stall-safe. While this can certainly be true, stalling of the canard surface must be expected to have consequences. As the history of airplane design demonstrates, therefore, canard configurations have only been built and tried (23,a, b) for experimental purposes. There are specific applications, however, particularly in winged missiles, where operation at higher lift coefficients may not be required. There can also be other inducements in the areas of drag and ar­rangement VTOL types (23,f) in favor of a canard system, such as in large supersonic airplanes (24), for example.

Across the Span. The magnitude of downwash is primarily considered in or near the plane of symmetry. The distri­bution across the wing span depends on the lift coeffi­cient. In rectangular wings, downwash increases slightly on either side of the plane of symmetry. In tapered plan forms, it decreases on the other hand; and it does so to a considerable degree in highly tapered wings such as that in figure 30. As a consequence, the stabilizing effectiveness of the horizontal tail, is a function of its lateral location. Results when using a pair of surfaces attached to the nacelles, as in part (B) of the illustration, are as follows:

In short, the inboard panels are very inefficient by com­parison, on account of a high downwash ratio.

(23) Characteristics of canard-type airplanes:

a) Focke-Wulf “Ente”, see NACA Rpts AC59 (1927) & 132 (1931).

b) Hubner, Focke-Wulf F-19 “Ente”, ZFM 1933 p 223.

c) Kiel, Stability of “Ente” Airplanes, NACA TM 612 (1931).

d) Flying Model of Curtiss P-55, tested in Full-Scale Tunnel, NACA W Rpts L-627, 630, 650 (1943/45).

e) Sandahl, Body Wing Canard, NACA TN 1295 (1947).

f) Borst, High Speed VTOL Aircraft Aerospace Eng. Aug 1962

(24) Hibbard, Supersonic Transport Airplanes (Aero Space Engg, July 1959 p 32) with V 1500 kts, are proposed with Canard Control. Drag penalty associated with trim at M = 3 is so great, that canard configuration is selected.

Tandem Configuration. We will first consider the tandem system as in part (a) of figure 31. The separation distance between the two wings may be I = b. The upwash at the place of the forward wing is assumed to be zero, while the down wash at the second wing may correspond to dz/doC = — 0.5. Using equation (2) balance requires

Cm,=CLl (S, /S)(Щ) = |Cm2J

= CL2 (S2/S)(|^/i|) (34)

where S = S, + S2, and і = lt + {$2) = separation distance = reference length. In terms of the angle of attack

(dCL / do: )oc (S f /S)( і, /І) = (dC L /do:) (осц +i)(S2/S)|^/

(35)

where <Хц = (1 — 0.5) ос, and і = angle at which the second wing is set against the first one. One and the same slope dCL /doc shall be assumed to apply to either wing (when flying alone). Therefore:

ocS, i, =(0.5 a +i)SeUz) (36)

where the absolute value is used for the агпъ?2 . In a tandem system (where S( = S2 = 0.5 S):

ос Лх = (0.5 ос + i/oOI-41 і or -4 = + l!°c) 1

(37)

where the 0.5 reflects the influence of the downwash.

Stability. Using next equation (5) static stability requires that

I dC^/dof I ^dC^/dct’

0.5(dCL /da: )(S2 /S) I / Д |>(dC L /doc )(S, /S)(І Ц)

(38)

Note that Jt is considered to be positive, and £ to be negative. For neutral stability, thus:

0.5SU | = S,^

(39)

and in a tandem system (where 82=8, = 0.5 S)

0.25li2l = 0.5^; or і,=0.5|ігІ (40)

In fact, to be stable, must be somewhat shorter than 0.5 Jz. Introducing now the maximum permissible J{ into equation (37)

we correctly obtain і = 0. We will now specify a stability corresponding to dCm/dCL = (dC^/da: )(da: /dC L), for example = — 0.01, which means a stabilized length of д хЦ = 1% of the separation distance For an assumed dCL jdoC = 0.06, we then obtain

dCfJdoC = – 0.06 (д хЦ) = dC^H/doC + dC^JdoC

= 0.06 (0.5) (JJJ)- 0.5 (0.06) 0.51 Je IЛ I

= 0.03 — 0.015 І і / J I

Since I JzI = J-J, , we obtain

dCm/doC° = 0.03 Ц /I) – 0.015 (1 – і, /І) – 0.06 (д хЦ)

and the moment arm of the first wing:

£il£ = (1/3) — (4/3) A xjj (42)

For neutral stability (kx/J = zero), this equation cor­rectly yields =1/3.

Control. Combining equations (37) and (42)

Jji = (0.5 + i/oc) J£ll = (1/3) – (Ф)(АхЦ) (43)

B) CANARD-TYPE d = ZERO Sc = 0.2 S

After replacing again, we eventually obtain the ap­proximation

i/oc = – 6(д x/J) + 8(д xUfsz – 6 (д x//> (44)

Thus, the tandem system considered can be expected to be stable when and if the CG is ahead of the 1/3 point of the separation distance. The configuration can be con­trolled or trimmed by reducing the angle of attack of the rear wing, roughly at the ratio of ді/д ос = – 6% for each stabilized percent of the separation distance.

Canard Analysis. Assume that the forward wing reduces in area, say to 25% the rear wing, or to 20% of the combined area. In a xough analysis, we may assume no interference between the two “wings” (no downwash). The neutral location of the CG is then found to be at

i, = li2KS/S ) = 510,1 = (5/6)^ (45)

where Sc = area of the canard surface. To be stable, the CG must be placed 1% of the reference length ahead of the neutral point, for each increment equal to 0.01, of the stability derivative dCryydCL. This is as in conventional airplane configurations, but only when the wing is not affected by downwash or upwash, respectively of the smaller, stabilizing surface (located either ahead or aft).

dC^dc* = +0.015 FUSELAGE

= +0.048 F + CANASD = -0.037 F + WING dCL/dc£ = 0.070 WING + F

Basic Configuration. The large-scale model of a canard configuration is shown in figure 32, consisting of a plain body of revolution, a straight wing in high position, and a stabilizing surface mounted mid-wing near the nose of the fuselage. Pressure-distribution tests indicate that the load distributions across the span, both of the wing and canard surface, display considerable dents in the center. The one in the wing, seems to be responsible for an induced drag at least 10% higher than expected. The effectiveness of the canard surface can be understood when assuming that the part of the fuselage “covered”, only produced 1/2 of the lift as it would, without the fuselage present.

Canard Effectiveness. Investigation of the canard surface of light weight Curtiss P-55, figure 33 airplane in the NACA’s Full-Scale Tunnel (23,d) reveals:

(a) When rotating the canard surface against the fuselage, its force varies corresponding to dC ^ /di = 0.042.

(b) When rotating the surface together with the fuselage, the differential corresponds to dC ^н/dor = 0.057.

The load distribution, plotted in figure 34, shows a deep dent or gap in the center of the canard-surface span. The pressure distribution induced upon the fuselage, does not fill up the gap. Evidently, there is not much body left ahead of the leading edge, where a cross flow could develop. In fact, the gap is so big that we may consider 15% of the canard surface to be cut out and to be ineffective. Using the function explained in the Chapter III the “lift angle” may thus be expected to be

dar/dCLH = (10.5 + 19/AH )/0.85 =

14.9/0.85= 17.4° (46)

where Au = 4.4, including the part covered by the fuse­lage. The result as under (b) above is 1/0.057 = 17.6°. Note that we did not reduce the effective aspect ratio since the boundary layer and/or wake at the nose of the fuselage are very small. Note also that area and aspect ratio and lift curve slope of the canard surface used in the following tests, are larger than those of the surface used in figure 34.

Stability Contributions. The pitching moment char­acteristics of the P-55, as tested with the CG located at 12% of the “mac” (id est, behind its leading “edge”) are plotted in part (a) of figure 33. Without the stabilizer, the configuration is stable, dCm/doc = – 0.0063 or dC„,/dCL = – 0.0063/0.07 = – 0.09, where 0.07 = dCL/doc as tested. To balance this moment at a certain lift coefficient the canard surface has to provide a positive (destabilizing) moment. Stabilization in any canard configuration can thus only be obtained from the wing and that contri­bution is not adequate in the condition as tested, with the CG at 12% “mac”. Positive stability can be obtained, for

example, by moving the CG forward to the leading edge of the “mac”. Part (b) of the illustration shows the corresponding pitching moment characteristics. To trim the airplane, say to CL = 0.5 where oc = 7 , a stabilizer angle і of at least + 7 is required. The aerodynamic angle is then осц = оГ+і = 7 + 7=14, and the lift or normal-force coefficient of the canard surface is between 0.8 and 0.9. In other words, this coefficient grows faster than that of the wing.

Stalling of the canard surface takes place at oC ^15°, where Сиц ~ 1.0. Stability is still preserved at the corresponding lift coefficient of the wing (CL ^ 0.6). In fact, stability increases on account of stalling. It is com­pletely impossible, however, to trim the airplane above this limiting coefficient. Since negative lift in the canard surface will “never” be of any practical interest, the airfoil section used should preferably be cambered, thus pro­viding higher maximum lift. To avoid sudden stalling (which might prove to be “fatal” for the airplane) the section shape should be such that stalling starts gently (from the trailing edge). One way of extending the range of effectiveness of the canard surface, is to make its surface larger. Roughly, to double the maximum lift co­efficient at which the P-55 could be flown (from CL = 0.6 to 1.2) it would be necessary to double the size of the stabilizer area (from 10% to 20% of the wing area).

Aspect Ratio. It is basically desirable that the canard foil produces comparatively large positive lift-forces — with a lift-curve slope as low as practicable. Therefore, the ca­nard surface should have a small aspect ratio. For ex­ample, the “larger” surface (S H = 10% S) used in the configuration in figure 33, has a geometrical A = 6.8. Upon reducing this ratio, say to 1/3, its lift curve slope can be expected to be reduced to 70%. As a consequence, stability would be increased by Д (dC^dC^) = – 0.06. After relocating the CG, accordingly, control (trim) might then be obtained up to CL ~ 0.8 (instead of 0.5 as with the large-aspect-ratio canard surface).

THE LARGE FIN, FOUND NOT TO BE SUFFICIENT, WAS AUGMENTED BY A PAIR OF AUXILIARY FINS MOUNTED BELOW THE WING, NEAR THE TIPS.

Figure 35. Focke-Wulf F-19 “Ente” research airplane, built and flight-tested (23,a, b) between 1927 and in 1936.

Full Scale Experience. One of the few canard-type air­planes built and flight-tested is the Focke-Wulf “Ente” (first 1927, and then 1936), shown in figure 35. Reports by its designer and the principal research pilot (23,a, b) reveal practical results as follows:

A) Advantages:

(a) It is true that the wing of a canard-type airplane cannot be stalled.

(b) The canard type does not have any slipstream interference.

(c) Lateral (aileron) control is fully available, at all times.

(d) Even when and if the airplane could be controlled to angles of attack above stalling (as tested to (X = 45 , in a wind tunnel) spinning is “safe”; the CG located “ahead” of the wing prevents flat spins.

(e) The configuration lends itself to the use of a nose – wheel tricycle landing gear.

B) Disadvantages:

(a) When the canard surface stalls, the airplane re­covers by way of a certain dive. This is dangerous when it happens near the ground and in particular during the landing approach.

(b) The fuselage (extending far forward) is destabi­lizing about the vertical or normal axis. As a con­sequence, unusually large vertical “tail” surfaces are required.

(c) Longitudinal control is sluggish by comparison, particularly during take-off where the slipstream does not help.

(d) In regard to speed performance, the canard con­figuration is said to be potentially inefficient.

In view of the positive load to be carried by the canard surface, the 1936 “Ente” had an area ratio S^/S= 17%, a well-cambered section, set at an angle of і = + 10 , and it is equipped with full-span trailing-edge flaps, used as ele­vators. At a usual CG position 26% “mac” ahead of the wing’s hedge, the stability corresponds to dC^JdC^ = — 0.10. When taking off at C L = 1.2 (total on wing surface), the lift coefficient in the canard surface must have been CLH between 1 and 2.

High-Speed Configurations. While the canard configu­ration does not seem to be profitable when used in other­wise conventional airplanes, there may be other appli­cations where stabilization by means of a surface ahead of the wing will be desirable:

(a) in winged and guided missiles, where higher lift coeffi­cients (as in airplanes when landing) may not be re­quired.

(b) in supersonic and low-aspect ratio airplanes, where the canard arrangement is said to present less drag.

(c) in configurations where the engines and their weight are located aft, a comparatively long portion of the fuselage is ahead of the wing. The fuselage can, there­fore, conveniently be utilized as a “boom” supporting the horizontal control surface.

Supersonic Airplane. Figure 36 presents the pitching mo­ments of a straight-wing configuration, tested at M = 0.7. As explained elsewhere, lifting and longitudinal charac­teristics at this M’number, are essentially or qualitatively as in incompressible fluid flow. From figure 36 it is noted that:

(a) Because of the low aspect ratios of 3.0 and 3.1, the CL(oc) functions of wing and control surface are slightly non-linear.

(b) The wing pitching moment non-linearity is increased by the swept-back position of the sharp lateral edges, in reference to the CG.

(c) After including a certain non-linear component due to the fuselage, the C^Cl.) function of wing plus fuse­lage, is distinctly non-linear. This is to say, the value of the negative grows with Cu at an increasing rate.

(d) After adding the positive moment due to the canard surface, stability is reduced to zero at near zero, for a position of the CG at 3% of the “mac”. The pitching moment coefficient increases roughly as Cm== -.0.2 CL2.

(e) The canard foil stalls at ос ц =10°, where its lift coefficient is somewhat above 0.5 and the moment contribution CmH between 0.07 and 0.08.

To Improve the longitudinal characteristics, the non – linearities characteristics of the wing (not the canard sur­face) should be eliminated, the maximum lift of the canard surface should be increased and its lift-curve slope be reduced. Steps as follows would be suitable:

1) reduce the canard A’ratio, and increase its area at the same time,

2) increase the canard moment arm, thus assisting area,

3) reduce sweepback (making leading edge straight),

4) reduce the fuselage diameter, if possible.

(25) Characteristics of high-speed canard designs:

a) Peterson, With Straight-Wing, NACA RM A57K27.

b) Boyd, Delta Configurations, RM A57J15 and A57K14.

c) Peterson, Delta at High Speeds, RM A57K26.

d) For comparison see same wing-body combinations as in (a),(b),(c), with horizontal tail aft in (19).

Fuselage Length. Comparative tests on the delta-wing con­figuration as in figure 37, show the influence of the length of fuselage and canard moment arm:

(a) The lift in the exposed panels of the canard surface is independent of fuselage length; ДС^/Ло^ = 0.063, based on exposed area (equal to 6.9% of that of the wing).

(b) The canard moment arms arejl4 = 1.2 c for the short, and 1.8 c for the long fuselage; the ratio is ~1.5. By comparison, the moment differentials due to the sur­face correspond to a ratio of 1.8.

(c) The longer fuselage causes an increment A(dCm/dCL) between -1 0.03 and + 0.04.

(d) The canard surface stalls at an angle of attack (oc + i) ~ 25°, where its lift or normal-force coefficient is in the order of 1.4. The corresponding maximum canard contribution is analytically:

Сти = + 1*4 (0.069) 1.2 ~ + 0.12 for the short fuselage
= + 1.4 (0.069) 1.8 ~ + 0.18 for the long fuselage

As discussed in the next paragraph, the “long” moment is even larger than this. The longer fuselage thus permits either to trim to at least 1.5 times the wing lift coeffi­cient, or to compensate for at least 1.5 times in terms of ^(dC^/dCL) of the configuration.

Interference. The canard surface leaves behind a certain downwash — and a pair of tip vortices. The possible interference of this system with the wing is mentioned in several reports (25). In the case where the canard surface is large the downwash is highly destabilizing. However if the distance is large and the front surface small the conse­quences of such interference are of minor importance. Theory predicts that the two lifting surfaces have the same induced characteristics as a wing to whose load distribution the lift of the canard surface is added. In reality, the wing load is expected to be reduced in the center, and to be increased outside the canard span. Again, this may not be important in a small straight wing. However, in the example of a delta wing as in figure 37, a relocation of lift from the center outboard, can be ex­pected to make the wing’s pitching moment more negative (nose-down and stabilizing). There is some indication of possible interference in (b) above, where the canard mo­ment grows to “1.8”, while the moment arm ratio is only 1.5. It can be speculated that the short-fuselage design exhibits a differential due to interference in the order of &(dCm/dCL) = – 0.02. This differential can readily be explained, as above. The fact that the long configuration does not experience a similar differential, may be asso­ciated with the much longer distance that the downwash field has to proceed, before it reaches the wing. It. is also likely that the canard downwash reduces the positive moment of the fuselage. Finally, de/doc at the location of the wing, must be expected to reduce, as the angle of attack is increased.

Delta Canard While above, the moment arm of the canard surface is found to be of value in postponing stalling, the fact remains that this surface always contributes a destabi­lizing moment. Using equations (38) and (44), the loca­tion of the neutral point behind the canard surface is found to be

– (S/Sc)/(1 + S/Sc)(dC L /doC )/(dC L /dot) (47)

Stability can thus be increased by reducing the lift-curve slope of the canard, in comparison to that of the wing. In view of

doc /dCL w 11 + 20/A

(Chapter III) this can be done by reducing the aspect ratio of the canard surface. When, for example, a wing with A = 6 is combined with a canard surface having Aw =6, equation (44) indicates IjJ = 5/6. Reducing Ац , say to 2, we obtain the ratio

(dCL/doO/(dCL/doc)„ =

(doc/dCL )й /(doc/dCL ) – 1.5 and a neutral point at

Jjf = 7.5/8.5 = 0.88

Using the various components of longitudinal moment together with the downwash mechanism, it is now possi­ble to evaluate or to predict the static stability of plain wing-fuselage-tail configurations.

Moment Due to Tail Figure 15 presents characteristics of a simple airplane configuration. The stabilizing moment derivative (dC^dCY ) of the horizontal tail is found from the difference of the tests with tail on and tail off. The tail’s moment is theoretically

dCmJdCL= – (dCL /doc) H (SH/S)(^/c)(l +df/d(r)(do:/dCL)

(27)

In the example as in the illustration, the known terms in this equation are dCL/doc = 0.07, Su/S = 0.15,4 M/c = 3.4, so that VM = 0.5. By comparison of the stabilizing mo­ments, with and without the wing, we find (1 + (dt/doC)) = 0.45, so that dt/doc = — 0.55. The rest is as tested.

IN THE GRAPH INDICATING:

A – AC^ = -0.09 CL DUE TO CG 9% AHEAD OF AC В – £Cm = -0.09 CL2 ШЕ TO CG 0.4 c BELOW AC C – <3Cm~ CL2 DUE TO VARIATION OF DOWNWASH df/dc<

Neutral Point. When moving the CG of an airplane aft stability reduces to zero when reaching the neutral point (neutral stability). For a given configuration, with a pitch­ing moment as in equation (27), this point is indicated by

(a/c)-(n/c) = dQn« /dCL (28)

To be stable, n must be longer than a (see figure 1). Since іц changes little when changing g and n, if the variation of the combined CL due to a change in CLW is small, the absolute value of the combined d/Cm/dCL (about the leading edge) is equal to n/c. The location “n” of the neutral point is thus aft of the leading edge. Usually, the fuselage contributes a positive Д (dCry/dCY), resulting in a forward shift of the AC while the horizontal tail provides the stabilizing dCniH/dCL, usually between — 0.1 and — 0.2, thus extending the neutral point, say to 30 or possi­bly 35% of the chord. All these variations are indicated by

Дх/с = – A(dC*/dCL) (29)

High Wing. Figure 24 gives an example for the neutral point of a “parasol” configuration. Between CL = 0 and 0.9, the neutral point moves from x/c = 0.34 to above 0.6. This variation is primarily caused by the stabilizing action of the wing’s lift at its AC in reference to the CG. In a manner similar to that in airfoil sections and/or in wings, an aerodynamic center can also be determined for an airplane configuration; the neutral point. This can ap­proximately by considering [105]

-2 C 02 04 Об 08 Ю CL

Figure 24. Longitudinal characteristics of a “parasol” type air­plane configuration. Aerodynamic center and neutral points, all including the horizontal tail, set at – 4.5 against wing zero lift line.

In the graph indicating:

A – С*, = ~ 0.09 CL due to CG 9%c ahead of AC В – Cm = – 0.09 CL2 due to CG 0.4 c below AC C – due to variation of downwash d/daf

where z = positive in a high-wing arrangement. Note that the influence of vertical displacement corresponds to that of a mechanical pendulum, the CG is suspended below the wing’s center of lift. It should also be noted that d(ACrr)/dCL = constant for a certain displacement “z” of the wing against the CG or vice versa. In the case of figure 24, it was found by trial and error, that the Cm(CL) function can be made into a line practically straight be­tween Cl = 0.2 and + 0.8, when raising the reference point to the level of z = 0.24 c, above the wing’s гею lift line. The neutral point NP (aerodynamic center of the configuration, including the horizontal tail) is thus found to be at

x = 0.34 c; and z = 0.24 c

measured from wing leading edge and above its zero lift line (which corresponds to oc0 = — 0.7 ). The explanation for the comparatively high location of the NP (in figure 24) may be found:

(a) in lift (and drag) originating and/or increasing in the rear end of the low aspect ratio “square” fuselage, as the angle of attack is increased;

(b) possibly of some progressive viscous interference of the fuselage with the wing’s downwash;

(c) in the low position of the horizontal tail in relation to the wing and its downwash sheet (see later under “low horizontal tail”).

It is thus seen that the downwash and its variation, as the tail moves up or down as a function of CL, contribute to the position of the “aerodynamic^center.” In view of the reduction of stability due to propeller and slipstream action (discussed later) an increase of dCrrJdCL as indi­cated in equation 30 may be desirable to a degree. On the other hand, strong stability at high lift coefficients, makes control of the airplane during a landing maneuver diffi­cult, particularly when the airplane is designed for tail – down and tail-first type of contact with the ground

Wing Wake. As shown in (15,d) the turbulent viscous wake emanating from the trailing edge of an average plain wing section, has the following characteristics at a con­ventional location of the horizontal tail:

x = 2 c, measured from trailing edge of wing

0/c = 0.5 Cds~0.005 = momentum thickness ratio

6л ~ 1.10 = displacement thickness of wake Vmin ~ 0-94 V = minimum speed, in center of wake S ^ 0.1 f c = total thickness of wake [106]

Depending upon the geometrical arrangement, the wake sheet may meet the horizontal tail surface, at a certain angle of attack or lift coefficient. A theoretical and ex­perimental study (15,a) indicates that in this event, the lift of the tail surface basically corresponds to the reduced dynamic pressure within the wake. However, within a field of flow with a velocity gradient in the direction normal to its span, an airfoil becomes attracted toward
the side of increasing velocity. The result can be a certain irregularity in the forces or moments supplied by the horizontal tail surface. Pitching moment coefficients of a simple airplane configuration are plotted in figure 25:

(a) for wing plus fuselage, mid-wing configuration,

(b) with wing and CG in high position

(c) for a symmetrical mid-wing combination,

(d) for a wing and CG in lowered position,

(e) for a wing and tail setting of + 4° against the fuselage.

By reasons of symmetry, configuration (b) must have an inflection around C L and/or oc = zero. There are evi­dently “dents” in the C^CjJ functions of (d) and (e), at CL ~ 0.3 and 0.4, respectively. At these critical lift coefficients, the slope dCm/dCL is locally reduced. Of course, in the operation of an airplane, the “mixing” action of the propeller (if any) or variations of wing shape causing a distortion of the vortex sheet (such as dihedral, cut-outs, flaps) may render the dent in the func­

tion insignificant.

Vertical Position of Tail There is considerable evidence indicating that an improvement of stability does take place, as the horizontal tail moves below the vortex and wake sheet. For example, the midwing configuration in figure 25 shows an increment of the tail-moment con­tribution at CL = 1, in the order of 8% (not counting the increment due to fuselage). The high wing or low-tail arrangement m the same graph has a tail moment of 12%.

Another advantage of the latter configuration (with the CG at the wing’s quarter point) is the fact that the horizontal tail is always below the vortex sheet where (as in figure 20) the downwash is less than the maximum. As a consequence, the low-tail configuration exhibits a dCm/dCL = 0.18 (at CL = 0.8), while the mid-wing type has only — 0.16. Results of a systematic investigation (20,a) are plotted in figure 26 and indicate:

(a) With the horizontal tail either above or below the vortex sheet, the stabilizing contribution of the tail is increased in comparison to that on fuselage center line.

(b) The improvement is greater below than above the wing chord. The derivative d£/dz is evidently favorable be­low, and unfavorable above the vortex sheet.

(c) Directly on the fuselage (at its center line) the tail contribution is extremely low. It must be assumed that viscous interference (described in the “horizontal tail” section) is maximum here.

‘T” Tail. An extreme measure to avoid interference with the downwash sheet as well as with any wake, is to place the horizontal tail as high as possible. An example of a very high location of the horizontal tail surface, on top of the vertical surface (thus forming a “T” tail) is shown in figure 27. The tested pitching-moment derivatives (at lift coefficients below ~ 0.5) are as follows:
where do^ц /doC = (15 — 11.5)/15 = 0.23 only. Using the ratios as indicated in the illustration, and the lift angle of the wing doc /dCL = 15 as tested, the value dCmH/dCL~

— 0.085 is obtained. This compares with the test value of

– .08 shown on figure 27. Considering now the horizontal

HORIZONTAL TAIL CONTRIBUTION:

dCmH/dCL Лх/5 ON FUSELAGE CL -0.08 +8%

MT" TAIL POSITION -0.25 +25%

where the members in the last column indicate the stabi­lizing differentials of the horizontal tail. The + 0.13 is essentially due to the fuselage, hinged at 0.54 of its length (a point not typical in conventional airplanes). With the horizontal tail located on the fuselage center line, the configuration is not stable. Analysis using equations 22 and 23 suggests that the downwash angle is

de/dCL ~ — 9.2 — 2.3 = — 11.5°

which is 2.5 times the induced angle of attack. The influence of the fuselage upon the effective aspect ratio is estimated to result in AL = 0.7 (4) = 2.8. Using the corresponding lift angle of

(doc/dCL)M = 10.5 + 6.5= 17°

the contribution of the horizontal tail then is calculated to be

^mn/dCL – — (doCM /d<X ) (SJS) (4 /с) (da: /dCL )/(doC /dCL )Figure 27. Influence of vertical position of the horizontal tail

(32) surface (20,d) on longitudinal stability. Rc = 8(10)6.

surface raised to the top of the vertical tail, we may expect that the downwash is reduced to 0.6 of that assumed above (see figure 20). Fuselage interference is assumed to be zero; in fact, the upper side of the hori­zontal surface is completely undisturbed. Since AH = 4 = A as in the wing, we obtain dotCM/dc*r = (15 — 7)/15 = 0.53, which is more than twice the 0.23 ratio for docH/dcxT above. Consequently:

dCmw/dCL = – 0.53 (0.2) 2.1 = – 0.22

while – 0.25 was obtained from test (see figure 27). The “T” tail configuration thus does two things:

it avoids the downwash maximum near the vortex sheet,

it avoids viscosity-induced fuselage (wake) interference.

“71” Tail at High Angles: While the “T” tail, as in figure 27, is safely placed outside (above) the wing wake, the variation of downwash as in figure 20, should be expected to reduce stability, in the range of higher lift coefficients corresponding to what is explained above under “low horizontal tail”. Indeed, comparison of the C^Cl) func­tions in figure 27, without tail and with “T” tail, reveals a reduction of the stabilizing tail moment between CL = 0.3 and 0.6, similar in absolute magnitude to the stabilizing amount of I 0.45 I, derived above for a low-tail con­figuration. Therefore, the advantage of the “T” tail ar­rangement rests primarily in the elimination of fuselage interference. Disregarding structural considerations, a really high location of the horizontal tail has another disadvantage. At very high angfes of attack, the horizontal surface is bound to get into the wake from the stalled wing. As seen in figure 27, the pitching moment then turns unstable (into the positive direction) thus presenting a situation which is likely to get into the “deep stall condition, Chapter XVI.

Aspect Ratio. A given tail plus fuselage assembly was tested (17,f) in combination with 3 wings having the same area, but differing in aspect ratio, between 2 and 6. The lift of the horizontal tail is found as the differential between the complete configuration and wing plus fuse­lage (tested without tail). Including for A = go, or 1/A = 0, the slope (dCL/do^ )й of tail plus fuselage tested alone, figure 28 demonstrates how the lift of the tail is reduced with a reduction of aspect ratio on account of increased downwash. It follows that

doCH/da: = (dCLH/doe)/(dCL/da: ) (33)

and del doc = doc jdoc – 1, so that

de/dCL = (docH/doc)(doc /dCL) – doc /dCL

where doc/dCL = lift angle of the respective wings as tested (plotted at the top of the graph). It is thus seen

that down wash increases and the stabilizing contribution of the tail reduces as the aspect ratio of the wing is reduced. For example, at A = 2, where ijc = 0.37 //1.08 = 1.3 and V =1.3 (0.2) = 0.26, the tail contribution is dCmw/dCL = – (dCLH/doO Vw (d(X/dCL) = – 0.03 (0.26) 25 = — 6.5%, which is only ~ 1/3 of what it is in average conventional airplane configurations.

Big Fuselages. Small aspect ratios are used on winged missiles (21), where not only down wash is a problem but also the size of the body in comparison to wing and tail is critical. Another example involving small wings and a large fuselage, is the high-altitude research craft X-15. Charac­teristics of a model similar to this airplane are presented in figure 29. The maximum width of the body is 0.34 b; its plan-form area is — 1A times total wing area. Analysis of these data show:

(a) The lift curve slope (without horizontal tail) within the range of small angles of attack, dCL/doC = 0.055, corresponds to A = 2.5.

(b) The maximum lift coefficient (without tail) is CL~ 0.7 CM= 1.5, at oL =45°.

(c) When adding the horizontal tail, its contributions to lift and pitching moment are not linear.

(d) There is a critical angle of attack near 50 , where forces and moments exhibit discontinuities.

The non-linearities (c) and the discontinuities seem to be a consequence of the fuselage flow pattern (separation). It must be noted, however, that the wing eliminates most of the fuselage lift, by way of downwash, so that the lift- curve slope as in (a), is fairly constant.

Tail Contributions. The interaction between fuselage and horizontal tail in the configuration as in figure 29, is complex. A qualitative analysis is as follows:

(a) By comparison of dC^/di = — 0.026 with dCL/di = 0.020, a tail length = (0.027/0.020)c = 1.35 c is found.

(b) Based on exposed area of S ц = 0.26 S, the lift-curve slope of the tail is dCLH/di = 0.020/0.26 = 0.077, which seems to be proper for an “exposed” aspect ratio of A = Ъгц /S ц = 6.3.

(16) Simple aiqilane configurations in wind tunnels:

a) Schlichting, Interference, Ybk D Lufo 1943, Rpt IA028; see (10,e).

b) Gruenling, Investigation of Me-109 Model, ZWB UM 7857 (1944).

d) Sherman, 17 Combinations, NACA T Rpt 678 (1939).

e) Service Technique Aeronautique (Belgium), Bull 15, Feb 1935.

(17) f) Stivers, NACA TN 4238 (1958) and NASA D-13

(1959) .

g) Fournier, Three-Body Configuration, NASA TN D-217

(1960) .

(18) Influence of vertical CG location:

a) Andrews, Flight (Aircraft Eng) 1937 No. 1474 & 1479.

b) Francis, Notes on Stability, ARC RM 1833 (1938).

c) Bryant, “Puss Moth” High Wing, ARC RM 1687 (1935).

(19) Further variants of the configurations as in figures 25,26,27, references (17,f)(18,d)(20,d), are reported in NACA RM A52D23/E01/L15a, in TN 3551,3857, and in NASA D-13; between 1952 and 1959.

(c) At <X = 20° , the tail moment differential is A С mul A <X = — 0.022, in presence of the wing. This is less than the dCm/di = — 0.026 above.

(d) Again at oc = 20°, the lift differential due to tail is ACl/A<X – 0.016. Combining this value with the derivative in (c), a tail moment arm Лц = (0.022/0.016)c = 1.38 c is obtained.

The difference between the derivatives as agains: j and <x, is due to downwash as well as to a more or less unknown contribution of the fuselage. On the basis ofS*» (total) =

2.1 S ц (exposed) = 0.55 S, a (AC L/Да: ) и ~ 0.016/0.55 = 0.029 is obtained. Comparison with 0.077 as in (b) leads to doc н I doc = 0.029/0.077 = 0.38 and df/doc = – 0.62. These values seem to be reasonable. More accurate pre­dictions cannot be made, however, for a configuration:

where the fuselage is long and wide,
where the tail length is no longer than ^ wing chord.

To show the problematic character of the analysis, the downwash as derived above, is only 1.4 times the induced angle, which seems low.

X-15. The model in figure 29, is evidently one repre­senting the NASA-North American research airplane “X-15”. Flight-test reports (22,c) reveal that the airplane is flown approximately:

Long Fuselage. The configuration in figure 30 serves first as another illustration for a long fuselage (forward). As­suming that the body in the middle may have a weight equal to 1/4 of the total, its addition moves forward the combined CG more than 20% of the “mac”; thus A (dCrr/dC _ ) ~ — 0.20, on account of fuselage weight. Its aerodynamic contribution is, on the other hand, Д (dCm/dCu ) = + 0.07. A long fuselage forward thus more than cancels its destabilizing effect; and we have an ex­planation, how long-nosed configurations such as that in figure 29, for example, can be stable in longitudinal direc­tion.

Zero-Lift Moment. Theoretically the fuselage does not develop lift. Its free moment (+ 0.07) has to be com­pensated, however, by a positive lift force in the hori­zontal tail. At higher lift coefficients, say at C L = 1.0, therefore, АСтн ~ — 0.07, which means in the con­figuration considered a AClh = + 0.07/0.66 ~ 0.1 and a total CLH ~ 0.2. This would not be a problem; in fact, in an airplane using wing flaps, the fuselage moment would tend to overcome the negative (nose-down) moment pro­duced by those flaps. On the other hand for a range of small lift coefficients (at high speeds) near Cb = 0, the differential depends entirely upon the angle “i” at which the wing is set against the fuselage. In a convention design, і (measured at the wing’s zero-lift line) is in the order of

at c< = 10° and with і = – 12° launching

at а: = 5° and with і = – 9° climbing

at oc = 15 ° and with і = – 20° recovering

s

This means that operation is at C L between 0.2 and 0.8. This range is small in comparison to that in figure 29. Other data and expected performance (22) of this airplane are:

(20) Influence of vertical location of the H4a. il:

a) Stivers, Vertical Location of Tail, NACA RM A57I10.

b) Multhopp, “T” Tails, Aero Digest, May 1955 .

c) Interaction with V’tail, see NACA TN 1050 & 2907.

d) Letko, in connection with (17,c), NACA TN 3481 (1955).

(21) Nielsen, Wing Body Tail Combinations, NACA T Rpt 1307 (1957).

(22) Aerodynamic characteristics of the X-15 research airplane:

a) Clark, Space Flight, Aero Space Engg, April 1959 p 43.

b) Bowman, Tunnel-Tested Characteristics, NASA TN D-403 (1960).

c) Flight Test Analyses, NASA TN D-615,623,723,1057,1059,1060,1159 (1960/62).

d) Weil, Review of Operation, NASA TN D-1278 (1962).

f) For data, see Jane’s All the World’s Aircraft 1961/62 p 292.

Figure 30. Three-body airplane configuration tested (17,g) at M = 0.2 and Rc := 2(10)4*. See text for principle results.

several degrees, so that the fuselage may be in horizontal position when cruising. Thus і = 0.3/0.06 = 5 , for ex­ample, for an assumed dCL/d<Y = 0.06. To balance the fuselage moment at CL =0, for instance high speed dive, a negative (down) load in the horizontal tail is required. In the example considered, that load would be in the order of 50% of the airplane’s weight (estimated on the basis of conventional values for tail volume et cetera), assuming a terminal speed two times the cruising speed and disre­garding any changes due to compressibility.

Taper Ratio. It is explained in the “downwash” section, that rectangular wings have less, and tapered plan forms more downwash in their center plane. Using a final (rolled-up) ratio t/ocL = 2.6 and an increment of at least 0.1 (as derived from equation 23),

d£/doc = – (tlocL )(d(*L /doc) = – 2.7 (5/17) = 0.8

or even higher, can be estimated for the doubly-triangular wing shape in figure 30. Analysis of the wide and large tail surface (S ц/S = 0.4) stretched out between the two nacelles (with A H = 4, plus end-plate effect supplied by the vertical fins) leads to an average effective ratio o^h /ос somewhat below 0.2. The configuration shown, or any similar, highly tapered wing, is thus rather ineffi­cient in regard to longitudinal stability.

In its conventional location, the horizontal tail of an airplane is obviously exposed to the downwash produced by the wing. Realistic prediction of downwash is some­what complex, although the flow pattern can readily be understood.

Mechanism of Downwash. Considerable effort has been put into the analysis of downwash (11) as a function of:

a) wing plan-form shape and aspect ratio,

b) roll-up situation and longitudinal tail location, Figure 17. Horizontal plan, showing the geometry of vortex sheet

c) vertical location of the tail, in relation to the down – and hp-vortex Pair – wash sheet.

None of the solutions are complete, let alone simple enough to be used in practical design work and the agree­ment with experimental results is conditional. The mecha­nism of downwash can physically be explained, however. Consider the point (o) in figure 17, each of the lateral vortices emanating from the wing tips, induces a down – wash velocity at that point. The “bound” vortex (the lifting line) also induces a downwash component and this component directly representing the circulation around the wing sections, reduces as the point (o) considered is moved downstream.

(11) Analysis of wing downwash:

a) Flugge-Lotz, Report on Downwash w’out and with Slip­stream, Ybk D Lufo 1938 p 1-172; also Lufo 1938 p 552 & 1940 p 161.

b) There are over 100 other references listed in (a).

c) Kaden, Rolling-Up Process, Ing Archiv 1931 p 141.

d) Silverstein, Downwash & Wake, NACA T Rpt 648 (1939).

e) Diederich, Calculation of Downwash, NACA TN 2353 (1951).

f) Spreiter, Trailing Vortex Sheet, J Aeron Sci 1951 p 21.

g) Helmbold, Downwash Analysis, ZFM 1925 p 291 & 1927

p 11.

h) Multhopp, Downwash Analysis, Lufo 1938 p 463.

Trailing System. Figure 18 shows a lateral view (a) of the central part of the vortex sheet, originating from the trailing edge, and (b) of the pair of trailing vortices de­veloping from the wing tips. The vortex sheet leaves the trailing edge in a direction corresponding to the aero­dynamic angle of attack (measured against the zero lift line of the section). The maximum downwash angle is<fx = — oc0, accordingly. Behind a wing with elliptical distribu­tion, the downward velocity of the tip-vortex cores is

w = – (CL I A) V4/tT3 ; or w /V = <XL 4/тг’2 ^ 0.4 odd

(21)

where w and, both are negative quantities. The final velocity in the plane of symmetry is

w =-(CL/A)V(4/’rr); orw/V = (16/tf )o^l.6aT=£

The cores thus move down at 1/4 of the final downwash velocity in the plane of symmetry.

Downwash Due to Circulation. The component of down – wash representing the circulation around the bound vor­tex (lifting line) is not “permanent”. It affects the hori­zontal tail surface, nevertheless. For elliptical life distri­bution (along the wing span) the angle (in radians) due to circulation velocity “v” is basically

£,-= — v/V ~ — (0.25/Tf) C(_ (с/х) (22)

where x = downstream distance from the lifting line at the quarter chord point, in the center of the wing. In terms of the induced angle of attack <xi = — Cl/tTA,

£c/c^l = 0.25 (x/b) (23)

This function, plotted in the lower part of figure 19, reduces rapidly as the distance x – tail length) is in­creased. It is, therefore, desirable to locate the horizontal tail as far aft as possible in order to obtain good stability.

Rolling Up. Upon leaving the trailing edge of the wing, the vortex sheet immediately starts rolling up (11 ,c) at the lateral edges into a pair of tip vortices. Using terminology as in figure 17, the distance (12) behind the wing’s lifting line for roll up is for a wing with elliptical loading:

x/b = 0.28 А/С (24)

For example, at C ^ = 1 and for A = 5 (as in figure 18) the distance is x = 1.4 b. Thus, at the usual location of a horizontal tail, rolling up is not yet completed. After completion, the “sheet” has disappeared. In its place we have then a downwash “field” between the pair of trailing vortices.

Final Downwash. The magnitude of the downwash com­ponent resulting from the trailing vortices reduces as the roll-up process goes on. In particular, the angle ratio behind elliptical wings theoretically assumes the final value.

,£ /orL = (4/rrf= 1.62 ~ 1.6 (25)

Figure 18. Vortex sheet (in the plane of symmetry) and tip-vortex location as seen in the vertical plane. [104]

Along the center line of the wing, theory (11) indicates maximum values (found “in” the vortex sheet) in the form of a constant multiple of the induced angle of attack oc,; of the wing, approximately as follows :

In highly tapered (triangular) wings, lift and downwash are concentrated around the plane of symmetry. For wings with a rectangular plan form the downwash is relatively small in the plane of symmetry. Using equation (24) the downstream distance beyond which the “rolled – up” values tjocl are to be used is estimated using equation 24. No simple formulation for the transition from the “sheet” values to the final rolled-up levels are available however. Experiments confirm the influences of the lift coefficient, corresponding to the distance within which transition takes place. Results for a tapered wing (13,c) plotted in figure 19, are on the line roughly indicating an upper limit, for CL = 0.6 and coincide with a lower-limit line at CL = 1.5. Theoretically, this transition is a stabi­lizing component in the mechanism of longitudinal sta­bility of airplanes. It can be estimated to increase the tail-moment contribution by a few %.

Variation in Vertical Plane. Figure 19 is meant to show maximum values of the downwash angle as they are found at the level of the vortex sheet coming from the trailing edge of the wing. The vertical position of that sheet (see figure 18) is approximately given by an angle measured from the wing’s quarter chord point and against its zero lift axis. The value of this angle is always larger than that of the final downwash angle. For a rectangular wing of aspect ratio 6, at a distance x/b = 0.5, the angle is, for example, ~ 2 o£ . It is an empirical observation, however, that maximum downwash angles are found in a “layer” located somewhat above the viscous wake sheet. Above and below the “maximum” sheet, the angle reduces in the manner as shown in figure 20. No simple theoretical formulation of the reduction is available. The experi­mental results in the graph may be approximated, how­ever, by

d(A£/£x)/d|z/b| 1.5 (26)

where z = distance above or below the sheet. It can be stated that by placing the horizontal tail below the vortex sheet, longitudinal stability can be improved, particularly on account of the* downwash gradient along the path through which the tail of an airplane moves when chang­ing the angle of attack.

Figure 19. Maximum downwash angle ratio “in” the vortex sheet, in the wing plane of symmetry, as a function of downstream distance x (13).

(14) Influence of fuselage upon downwash:

a) Liess-Riegels, Analysis, Ybk D Lufo 1942 p 1-366.

b) White, Fuselage Interference, NACA T Rpt 482 (1934).

c) Schlichting, Wing in Interrupted Flow, Ybk D Lufo 1940 p 81.

□

Figure 20. Downwash distribution in vertical plane of symmetry, behind plain wings as tested (13).

Fuselage Downwash Interference. All downwash con­siderations so far, apply to isolated wings. With the addi­tion the fuselage in the form of a slender body the lift distribution changes across the span of the wing and thus a change of the average downwash at the tail is obtainable since

(a) As explained in context with figure 4, lift (and lift – curve slope) can be assumed to be increased at and near the wing roots with a corresponding increase of downwash (14,a).

(b) Viscous interference and/or flow separation along the wing roots may, on the other hand, reduce the down – wash.

(c) The fuselage leaves behind a viscous wake; downwash may be affected by this wake.

(d) As a consequence of cross or “2 oc ” flow, downwash can be expected to be reduced at the horizontal-tail roots.

The net result of fuselage interference thus depends upon shape and on quality of the wing-fuselage combination. For example, in figure 15, the tail effectiveness ratio (1 + dzjdoc) is reduced from 0.45 to (0.87/0.91)0.45 = 0.43, when adding fuselage plus tail in comparison to tail with­out fuselage. It can thus be concluded that the downwash ratio is increased from d^/doc = (1 — 0.45) = 0.55, to (1 – 0.43) = 0.57 or 4%, as a result of fuselage interference.

Since this is small a practical assumption might be that with conventional smooth wind-tunnel models the influ­ence of the fuselage upon the variation of downwash (de/doc ) might be disregarded, at least within the range of low and moderately high lift coefficients. There are, on the other hand, many examples, showing a reduction of downwash on account of viscous fuselage interference as discussed in following paragraph.

Wing Roots. Hardly any airplane design is as plain and smooth as, for example, the wind-tunnel model in figure 21. For instance, the real (if obsolete) airplane depicted in figure 22 (built before 1933) has a fuselage with rec­tangular cross-section. The lower edges of this shape lead into the leading edges of the wing roots. Ahead of each root, there is also an attachment supporting a pair of struts belonging to the landing gear. As a consequence, the flow along the wing roots is disturbed, if not separated at lift coefficients above 0.6. This disturbance causes:

(a) an increase, of the drag due to lift by ~ 30%, or 12% based on the total drag at CL = 0.7 (climbing).

(b) a reduction of horizontal tail effectiveness by 7% (viscous wake effect);

(c) a reduction of downwash (as shown qualitatively in the illustration) from an estimated ratio of dtjdoc = — 0.50 to-0.26.

While (b) is slightly destabilizing, the reduction of down – wash to ~ 1 /2, is stabilizing. Corresponding to A(dCm/dCL) = — 0.20 due to tail, the stabilized length is 20% of the chord, in the original rough condition of the airplane (with d£/d. oc = — 0.26).

Wing-Root Fillets. To prevent separation at higher angles of attack, particularly along the upper (suction) side of low-wing airplanes wing root fillets are used. With suitable fillets, the maximum lift is increased, buffeting in wing and tail assembly is avoided and climb performance is improved. However, when first introducing wing-root fil­lets (possibly around or before 1930) it was soon dis­covered that they tend to increase downwash and thus to reduce longitudinal stability. In the airplane, as in figure 22, addition of large fillets extending downstream 1.7 c, and to 0.25 c above the trailing edge, and 0.3 c in spanwise direction (at the trailing edge), produced the following results:

(a) CLX (without power) increased from 1.2 to 1.3,

(b) downwash increased from dz/doc = — 0.26 to – 0.55,

(c) stabilized length reduced from (0.20 to 0.13)c.

Derivatives of a simple, clean and smooth airplane con­figuration are listed in figure 21. The stabilizing effect of the tail, in the configuration without fillets, is 15% of the chord, and that of the tail without downwash is 32%. It follows that without fillets, doc^/dot = (1 + d£/doc) = 0.15/0.32 = 0.47 and de/dot = – 0.53. When adding fillets, the aerodynamic center of the wing plus fuselage combination is shifted to the rear, approximately 1% of the chord. Against the new AC point, the stabilizing effect of the horizontal tail is reduced corresponding to 2% of the chord. The influence of the fillets corresponding to this 2%, leads to (1 + de/dar) = 0.41 and to dt/doC = — 0.59. The downwash increase with fillets is 12%. And while this increment is not comparable to that in the extremely disturbed airplane in figure 22, the influence of wing-root fillets upon stability should not be disregarded.

Cut-Outs. Downwash can be reduced by pulling the rear ends of the fillets up. While this produces a distortion of the vortex sheet, it does basically not mean that the derivative of the average downwash angle de/doc would be reduced. An efficient reduction of the derivative, can “only” be obtained by reducing the lift-curve slope in the center of the wing. This can, and has been done, by disturbing the flow past the wing roots on purpose, by means of roughness or obstacles such as spanwise strips of material placed near the leading edge. Another method is to reduce the wing chord by cutting away from the trailing edge at the roots. An example of such a cut-out is presented in figure 23. Downwash at the location of the tail is locally reduced from dt/dC^ — 7 to a minimum of — 2 . For an assumed span of the horizontal tail equal to 1/3 of that of the wing, the average downwash angle is reduced to di/dCL — — 5 , which is 70% of that of the wing without cut-outs (and without fuselage). On the basis of the lift angle of the configuration dcx/dCL~ 13 , the downwash reduction obtained, means an increase of the tail effectiveness corresponding to (1 + dt/doc) from 0.46 to 0.62. Since cut-outs might cause increased induced drag, stability is seen to interfere with per­formance particularly during climb.

NOTE: WITH A IUSELAGE AND WITH LARGE WING-ROOT FILLETS (NO CUT-OUTS)

DOWNWASH ANGLES ARE fH 10% LARGER THAN LISTED FOR WING ALONE (NO CUT-OUTS).

Figure 23. Example (13,c) for the influence of wing-root cut-outs upon the downwash at the location of the tail.

The horizontal tail surface is a lifting device, similar to the wing and is appreciably affected by the fuselage to which it is usually attached.

Cross Flow. Fuselage interference upon the horizontal surface located in the conventional location “at” the tail is twofold. There is first the cross-flow effect mentioned above in connection with wing plus fuselage combina­tions. Figure 5 suggests, when placing a wing or a tail surface at or near the end of a fuselage, that the destabiliz­ing (positive) fuselage moment will be increased above its value in free flow (without the tail surface present). This is evidently a consequence of the “2or ” type upwash (5) ahead of the leading edge of the horizontal surface. If the fuselage’s destabilizing moment is increased, the stabiliz­ing effect of the body-tail combination is reduced, accord­ingly. For an average conventional configuration, the maximum increment A M/M^h ^ 0.5, as at xff~^ 1, in figure 5, can be found to be equivalent to a reduction of the stabilizing moment of the wings (larger than tail surfaces) represented in that graph and in the order of several percent.

Stalling. The above discussion regarding wing-fuselage in­terference, primarily applies at small and moderately high lift coefficients. At the higher lift coefficients, stalling in straight wings is likely to begin at the roots, where the flow is strained the most. The pitching moment changes, accordingly, in negative (stabilizing) direction. For ex­ample, in the wing-fuselage combinations tested in (7,b, c) values of in the order of — 0.1 or even — 0.2, can

be observed wnen stalling. The corresponding shift in the center of lift is between 8 and 16% of the wing chord (to the rear). Longitudinal stability is improved as a result of stalling (returning the airplane to lower angles of attack) a) because of the shift in the center of lift and b) from the reduced downwash in the center of the wing (along the fuselage). Fillets, designed to postpone wing-root stalling, will be discussed later, in connection with downwash. [103]

Fuselage Interference. As pointed out in “Fluid-Dynamic Drag” (9) when adding a fuselage to the horizontal sur­face, its lift distribution is usually disturbed (possibly interrupted) by the boundary layer of viscous wake de­veloping along the fuselage. To demonstrate this effect, we present first the situation as in figure 10, where a wing (or tail surface) is placed in a wind-tunnel stream, in the center of which the velocity distribution is “dented” (in, the direction normal to the span of the wing). After integrating the dynamic pressure across the span of the lifting surface, we obtain an average Aq/q which may also be considered to be a gap in the stream (normal to the surface) having a width bg/bH = Aq/q. The lift and/or the lift-curve slope of the wing is reduced on account of this gap in its center, not just in proportion to the average dynamic pressure, but at a rate increasing with the size of the gap. Qualitatively, the result in figure 10 can be explained:

a) when reducing the lift in proportion to the average dynamic pressure, thus accounting for a reduction of effective area corresponding to the “gap”;

b) when reducing the effective aspect ratio as a function of Aq/q in the manner as described in “Fluid-Dynamic Drag” (9).

5(10)5

0.8 m

The lower limit indicated in the graph, corresponds to A[ = 0.5 A (q/q), used in the equation (taken from the Chapter III)

do?/dCL = 11 + (20/Al) (15)

The experimental points drop from the upper line (cor­responding to (a) above) to that lower limit. The reduc­tions of tail efficiency thus obtained, are greater than those derived above under “cross flow”. A horizontal tail is basically smaller than the wings in figure 5; and the viscous type of fuselage interference is predominant, ac­cordingly.

The Load Distribution across the span of the horizontal tail surface of a “typical” fighter airplane was measured in flight (10Jh). Considering the cut-out for the rudder, it may not be surprising that the distribution shown in figure 11 has a considerable dent (deficiency) when sud­denly pulling up (thus producing a negative load). How­ever, the same type of dent is obtained after 2 seconds, when the airplane has reached a certain angle of attack, giving it zz constant acceleration (and constant lift) corre­sponding to a total of 3.6 “g”. For an elevator angle now near neutral, the tested lift coefficient CLH = + 0.5 is essentially due to angle of attack, thus including the cross-flow assistance by the fuselage (if there is any). The distribution obtained can be analyzed in two different ways;

a) We can assume that the airplane’s fuselage (including canopy, wing roots, engine parts, radiators, shanks of the idling propeller, and various spots of roughness) may have a drag corresponding to ACD= 0.01 (on wing area). Based on horizontal area Sw = 0.2 S, a coefficient CDH = 0.01/0.2 = 0.05 is then obtained. According to the evalua­tion in Chapter VIII of “Fluid-Dynamic Drag”, a ratio of Aj. /A = 0.6 is then found.

b) If completing the lift distribution in figure 11, across the span in elliptical form, a load can be estimated as follows:

L = 250 (тг/4) 12.8 = 2500 lb

where 250 = maximum lb/ft in the center, and 12.8 = bH. This load is 1.25 times as high as tested. To account for the difference, we will first assume a reduction of effec­tive area by 10%. Using the basic equation for the “lift angle” (see Chapter III)

doc/dCL= (11°/0.9) + 207(0.9 (AjA) 3.4) =12+6.5 /(AJA)

where 3.4 = Ац = geometrical aspect ratio including the part covered by the fuselage. Solving this equation twice (once for AjA = 1.0, and once for an estimated ratio in the vicinity of 0.6, as in (a) above) a ratio of A^ /А = 0.59 is found for the lift angle ratio of 1.25 estimated above. This result agrees with that in (a).

(10) Characteristics of horizontal tail surfaces:

a) Letko, Fuselage with Tail (23,c), NACA TN 3857 (1956).

b) Lyons, Lift of Control Surfaces, ARC RM 2308 (1950).

c) Greenberg, Free Control Analysis, NACA T Rpt 791 (1944).

d) Plate Interference, NACA TN 408 & W Rpt L-660.

e) For Airship Fins, see NACA T Rpts 394 and 604.

f) Engelhardt, Interference of Fuselage Upon Horizontal Tail, Aerody Lab TH Munchen Rpts 1 and 3/1943.

g) Gillis, Fuselage Plus Tail, NACA W Rpt L-391 (1942).

h) Garvin, Load Distribution in Flight, NACA TN 1483 (1947).

i) Polhamus, Stability with “V” Tail, NACA TN 1478 (1947).

Cmuw= where-MH = pitching-moment differ­

ential due to horizontal tail, this coefficient should theo­retically be equal to CLH. However the moments as tested (10,f) were some 5% less, which means that the effective tail arm is 5% shorter than the geometrical arm. This reduction, indicated in figure 12, is discussed later, under “effective tail length”.

Surface Roughness. To prove that the viscous wake of the fuselage has considerable influence upon the horizontal – tail effectiveness, some basic tests were made (10,,f). The stabilizing tail moment was determined both on a smooth and on a rather rough fuselage body. At angles of attack between 6 and 8°, differentials ACmw/Aod, taken from figure 12, are as follows:

The tail differential effectiveness is reduced 18% in pres­ence of the smooth fuselage (having a basic drag corre­sponding to CD = 0.05, or CDH = 0.02) and 36% for the rough fuselage (with CD = 0.20, or Сой = 0.08 ). Obvi­ously, the effectiveness reduces as a function of the fuse­lage drag, as explained in (9). If defining the coefficient

Figure 12. Example for the interference of a fuselage body upon the effectiveness of a horizontal tail surface as tested (10,f) RM = 4(10)5.

A) with plain and smooth fuselage; CD = 0.05

B) with sand roughness k/1 = 1/103; CD = 0.20 where CP based on S = d2/4.

Statistical Evaluation. In a typical (but smooth) configura­tion, the fuselage-tail interference may be such that the effective aspect ratio is reduced to AWf- ~ 0.7 AH. The corresponding reduction of a conventional tail’s lift-curve slope can be expected to be in the order of (1 — 0.7)/3 = 10%. Note that in this formulation, the viscous influence of the fuselage (usually expressed in form of a reduction of the dynamic pressure at the tail) is “fully” accounted for. In other words, we do not assume that the average dynamic pressure across the span of the horizontal tail, would be reduced in the order of 10%, simply because there is a wing ahead the wake sheet of which m ay not, or should not impinge upon the tail surface. Statistically, wetted surface and drag of the fuselage is proportional to (I or d or a)2, where Д = length, d = maximum diameter, and a as indicated in figure 13. At least for small a/b ratios and/or for smaller aspect ratios, it can then be shown that the reduction A(dCL /doc.) is proportional to (а/Ьц )2 . The corresponding theoretical function presents an upper limit to the experimental ratios on figure 13. Even the fins of airships (in the vicinity of (a/bH ) = 0.6— fit into the statistical pattern. Tentatively, for “smooth” fuselages, the ratio of lifting effectiveness, in comparison to that of the horizontal tail alone (without interference) is

T-ratio = (0.94 — 0.02) (1 – (а/Ьи f)

Fuselage Shape. Results of a wind-tunnel investigation (10,a) of a horizontal tail tested at the end of four fuselage bodies, differing in cross-section shape, are pre­sented in figure 14. Differentials taken between oc = 0 and 8 , based on horizontal tail area are listed in the illustration. The loss of lift in the presence of the fuselage (corresponding to the horizontal lift ratio”) is between 6 and 15%, caused by interference of plain and smooth fuselage bodies as described above. When added to the fuselage, the horizontal tail appears has the least inter­ference of lift and moment in combination with the “deep” and/or the circular shape. By comparison, the square shape seems to cause some disturbance at the roots of the horizontal tail, v/hile the flat shape although having the highest combined lift-curve slope, may suffer some­what from too much cross flow at the tail roots thus resulting in a differential ratio of only 0.85.

Figure 14. Influence of fuselage shape, and interaction with the horizontal tail surface, in regard to longitudinal stability (10,a).

Effective Tail Length. The stability, does not depend the lift produced by the horizontal tail surface, but rather on the pitching moment obtained from the equation.

dCm4/dCL = (dCLH /doc) (SH/S) a /с) (16)

In other words, the moment arm of the tail is important and that arm is not necessarily equal to the geometrical dimension measured to the 1/4 chord point. For example, in the first line of the tabulation in figure 14, the effective arm “x” corresponds to хЦц = 0.050 (0.36)/0.0199 := 0.91, where 0.36 = tail volume VH = (SH/S) CfH/c). The horizontal tail as tested on three of the fuselage shapes, produces moments corresponding to arms which are be­tween 6 and 13% shorter than the geometrical distances measured to the CG location of the airplane configuration for which the tests were undertaken. There is probably some mutual interaction, upwash (as explained under “cross flow” above) and viscous-type lift of the fuselage “at” the trailing end (see context to figure 7). Considering net stabilizing effect of tail plus fuselage (F + H) the “deep” shape in figure 14, is seen to be the most effective one (with ДСт//Лос°= – 0.0163). This cannot be a general conclusion, however. It seems that the arrange­ments as tested, with 53% of the fuselage length forward of the CG, is not typical. If, for example, selecting the 1/4 point of the fuselage length as reference or CG point, the flat fuselage configuration emerges as the most stable, with ЛСт/Лос ~ — 0.050 for (F + H), in comparison to — 0.040 as for the square shape, and ~ — 0.035 for the two other forms. Thus, to improve stability, it would be profitable to make the forebody “deep” or round (and short), while a flat or square (and long) afterbody can be expected further to contribute to longitudinal stability.

Horizontal Tail Contribution. Longitudinal characteristics of a simple airplane configuration are presented in figure 15. Evaluation of tail effectiveness leads to results as follows:

a) The loss of effectiveness of the horizontal tail due to fuselage interferences is 9%. The effectiveness ratio is 0.91, accordingly.

b) When adding the fuselage to the wing plus tail arrange­ment (as tested without fuselage), an effectiveness ratio of 0.023/0.0265 = 0.87 is obtained.

The fact that the tail moment under (b) is somewhat less than under (a) can be explained by way of increased downwash (to be discussed later).

I ~ | AH = 3.7; VH =0.15 (3.4) = 0.5

a) PERFORMANCE OK HORIZONTAL TAIL SURFACE:

CONFIGURATION TESTED (dCm,/dtf ) (dC^ /6) RATIO EXPLANATION

HORIZONTAL TAIL ALONE -0.023 -0.33 1.00 ON WING DIMENSIONS

MOUNTED ON FUSELAGE -0.021 -0.30 0.91 FUSELAGE INTERFERENCE

BEHIND WING (NO FUSELAGE)-0.011_ -0.15 0.45 IN WING DOWNWASH

IN COMPLETE CONFIGURATION -0.009_ -0.13_ 0.39 DOWNWASH & FUSELAGE

Figure 15. Longitudinal characteristics of a simple airplane con­figuration

wing: b/c = 5; 23012 section; Rc = 4(lo/ ; ref. (16,a)

fuselage: 1/d = 7; 1/b = 1.0; see also figure 6.

Free Elevator. In wind-tunnel investigations, longitudinal and/or other stabilities are usually tested with the control flaps (elevator) fixed. As explained in the Chapter IV, the lift-curve slope usually reduces in the “stick-free” condi­tion. Since it is desirable that an airplane remains stable for the stick free case the design and/or CG location should be such that a correspondingly reduced tail contri­bution can be tolerated. The reduction to be expected for an average conventional tail surface due to the stick free case may be in the order of 15%. However, tab-balanced control flaps can reduce tail effectiveness much more, while a large overhanging-nose type of balance may not reduce stability at all.

General Method. To obtain the stabilizing moment of the horizontal surface the following procedure is used:

a) Find in Chapters III & IX the lift-curve slope of the isolated tail as a function of aspect ratio, including such effects as roughness and control or balance gaps.

b) Determine the effective aspect ratio, reduced as a consequence of viscous fuselage interference, as ex­plained above (9), or use directly the effectiveness ratio as in figure 13.

c) Use in the determination of the pitching moment, an arm which is somewhat shorter than the geometrical arm, either by 10%, or by a differential equal to 1/2 the maximum fuselage diameter.

For conventional (more or less “round”) and completely smooth fuselage shapes, it may also be said that the horizontal tail together with the fuselage may produce almost the same lift as the tail surface alone. This seems to be true in a smooth-fuselage-tail configuration reported in (10 ,g). A similar result is found for the configuration as in figure 14, with the round fuselage shape. Then using an arm reduced, say by 10%, the right order of magnitude of the stabilizing moment may be obtained. Less horizontal tail contributions may apply in full-scale conditions, how­ever, as explained in connection with figure 11.

“V” Tail To reduce drag and weight the vertical and horizontal surfaces have been combined in form as shown in figure 16. As pointed out in Chapter III, the panels of such a “V” tail retain their normal force derivative dCfl /do^h where ocn = angle of attack measured in the direction normal to each panel’s axis. Since L = cos"!*1, and ocn := c* cosr, the lift curve slope of a “V” tail (when used in longitudinal direction) can be expected to be

(dCL/doOH = (dCM/d*n) cos2r (17)

where P = angle of dihedral, and the coefficients are based on combined panel area. For dCN/do^n the value of the same tail at Г = 0, is to be used, including fuselage interference. The plain configuration investigated in (10,i) exhibits essentially straight C^Cl ) functions within the range of lift coefficients below CL = 0.8, with CL* = 1.0. The corresponding derivatives are plotted in figure 16. The results are as follows:

a) To make an airplane stable, the folded-down “V” tail area has to be larger than that of a straight horizontal surface. On the basis of a conventional S^/S = 20%, a “V” tail area

Sv/S = 0.2 cos? r (18)

may thus be needed. In the design as in (10,i) a 41% tail surface ratio was used.

b) When varying the dihedral angle of the “V” surface, the derivative dCm/di (where і = angle against fuselage) changes corresponding to equation (17).

c) The axes about which the elevator flaps are deflected, have directions different from that of the angle of inci­dence. Therefore, as derived from “V” geometry, and as shown in figure 16, the derivative varies as

dCm/d6 ~cos T (19)

Sidewash. As explained in (10,i), a “V” tail surface is subjected, not only to downwash (in vertical or normal direction) but also to some sidewash induced by the wing’s lateral tip vortices. With the tail located above the vortex sheet, each panel of the tail in figure 16 receives sidewash in a direction toward the vertical plane of symmetry. As a consequence, their aerodynamic angle of attack (normal to their axis or edges) is increased. The

effective downwash angle (derived from the pitching mo­ment contribution) is reduced accordingly. As plotted in part (c) of figure 16, a reduction from dS/daf = – 0.33 to – 0.21 (id est by ^ 1/3) is thus possible. The stability of the airplane model tested, basically varies corresponding to the neurtral-point location

Дх/с: dCAnH/dCL/ —cos2r (20)

After modifying this function with (1 + d£/’doc) the variation as in part (d) of the illustration is correctly obtained.

Tail Size. Of course, when folded down toT1 = (), a tail surface with Sv/S = 41% (as in figure 16) is no longer suitable, and the location of the neutral point at 70% c would not make an efficient design. About 1/2 the area can be expected to be sufficient, while a vertical surface (in a size corresponding to Sv/S ~ 10%) will be needed. In conclusion, when using a combination “V” tail, it does not seem to be certain that any wetted area and skin – friction drag would be saved. There are other advantages, however, to a “V” tail. Therefore this type of tail will be discussed further, in connection with “multiple-engine” airplanes as well as in the Chapter XII dealing with “later­al stability”.

The basic components of an airplane are wing and fuse­lage. Their longitudinal characteristics influencing stability are presented here.

Stream Curvature. The longitudinal moment of airfoil sections are treated in the Chapter II as a function of the type and operating conditions. When using their charac­teristics, it must be realized that airplane wings, always having a finite span, impart a permanent downward de­flection upon a certain stream of air. Since the deflection takes place within the chord of the wing, the effective camber of its section is reduced. The corresponding reduc­tion of the usually negative free or zero-lift pitching moment coefficient is approximately:

A Cmo= + 0.044/A (7)

which means that for A = 4.4, for example, the aero­dynamic center is shifted forward 1% of the wing chord. Since most of the Cmo values presented for airfoil sections in Chapter II, stem from tests on wings with A ■= 6, the difference as per equation (7) is usually unimportant in the analysis of wings having conventional aspect ratios.

Aerodynamic Chord (3). Airplane wings are usually not rectangular accordingly their chord length varies along the span. A mean chord, to which pitching moments can be referred, should account for the distribution of lift (load) across the span. If possible, the mean chord should repre­sent the aerodynamic characteristics of the wing as to magnitude and location. This can be done for “straight” and untwisted elliptical plan forms:

9

5 = {2Ы)£~(c dy) = (S/3-гг) cx= 0.85 c* (8)

where s = b/s = half span, and cx = maximum chord (in the center). The location of the “mac” chord (equation 8) is at

5

y/s = (2kS)£ (cy/dy) = (4/Зтґ) s = 0.42 s (9)

Length and location of the chord indicated by these two equations, represents that of an imaginary equivalent “rec­tangular” wing. Therefore, when giving the wing some angle of dihedral, the (0.42 s) station can be expected to indicate the vertical location of the center of lift. How­ever, wings are rarely elliptical; and if they are, they may be twisted, they will be combined with a fuselage, and they may carry partial-span flaps or engine nacelles. In short, their lift distribution will not really be elliptical. Assuming, on the other hand, that the load distribution of a plain rectangular wing be elliptical, its mean aero­dynamic chord (equal to the geometric chord) would then be located at у = 0.42 s, while it actually is in the vicinity of 0.46 s.

Tapered Wings. The pitching moment characteristics of tapered wings as derived from their theoretical lift distri­bution, are tabulated in (3,c) as a function of taper and aspect ratio. The use of these results is not very practical, however, and is not always realistic. A better definition of length and location of the mean aerodynamic chord, re­flecting and approximating the findings in that report, and the result of equations (13) and (14) as well, is shown in figure 2. The lateral location of the “mac” varies between у = (l/3)(b/2) for zero taper ratio (as in doubly triangular or “diamond” wings) and у = 0.5 for rectangular plan forms. The chord length increases from c = (S/b) as for rectangular plan forms to (4/3)(S/b) for zero taper ratio.

(2) General treatment of longitudinal stability:

a) Braun, Rpt IB6 Ringbuch Luft Tech, 1940; also Lecture at AF Inst Tech, WP AF Base, Ohio, 1948.

b) Diehl, Engineering Aerodynamics, 1928 & 1936, Ronald Press.

c) Zimmerman, Analysis, NACA T Rpt 521 (193.5).

d) Gilruth, Prediction, NACA T Rpt 711 (1941).

e) Gates, Longitudinal Stability, ARC RM 1118 (1927).

f) Perkins-Hage, Performance Stability Control, New York 1950.

g) Babister, Aircraft Stability and Control, Pergamon Press, Yol. 1 1961

(3) Determination of aerodynamic chord and center:

a) Crean, A’Center, The Aircraft-Engineer, April 1936.

b) Diehl, Mean Aerodynamic Chord, NACA T Rpt 751 (1942).

c) Anderson, Tapered Wings, NACA T Rpt 572 (1936).

d) Lachmann, Tapered-Wing Characteristics, Flight, Oct. 1936.

e) Blenk, Stream Curvature, ZaMM 1925 p 36.

This “mac” definition, is generally used in all but the “old” reports of the NACA. However, in a wing with 0.5 taper ratio (as in the illustration) the mean aerodynamic chord is 1.02 S/b, and its location is at у = 0.45 s. The mean chord S/b at the position of 0.42 s (as in equation 9) might, therefore, as well be used in many practical wing shapes. The wing’s aerodynamic axis, connecting the aero­dynamic centers (or the quarter points) on the mean chords, moves up together with the two wing panels, when they are given dihedral.

Wing Moment. In conventional mid-wing and low-wing configurations (as in figure 1, for example), the CG is essentially at the level of the wing. In a typical high-wing or even “parasol” type arrangements, the wing can be at an appreciable distance above the CG. In such a case, the longitudinal force (primarily the tangential component of the lift) of the wing has to be included in the analysis. In reference to the CG, (the hinge axis about which the airplane is pitching) the wing forces produce the moment (marked by the subscript “g”):

Cmo= Cmo+ (ax/c)Cl + (z/c)CD5 – (z/c)(do: /dCL )Cl

(10)

where x = g — a — longitudinal distance between CG and the wing’s aerodynamic center (measured parallel to the zero lift line, positive when the AC is forward of the CG) and where z = vertical distance (positive when the AC is above the CG). As suggested before, the term containing the section drag coefficient may usually be disregarded. The “lift angle” in the last term is that in two-dimensional flow thus independent of the aspect ratio, theoretically equal to 0.5/rr and actually ^10° (nr/180) ~ 0.17 (in radians). This term represents the tangential, forward – directed and thus “negative” component of lift. Using (4) the lesser value 0.15, differentiation of the equation yields the stability contribution of the wing

dCm/dCL = (л x/c) – 0.3 (z/c) CL ; сІД CWdC* = 0.3 (z/c)

(11)

where 0.3 = 2(0.15). Thus with the airplane’s CG not at the level of the wing’s AC, dO^/dC^ can no longer be expected to be constant as a function of lift coefficient or flying speed. In case of a low-wing configuration (pro­vided that the CG is really above the mean aerodynamic chord of the wing) the wing’s dCm/dCL may grow less negative (less stable) as CL is increased, while in a high – wing airplane, the opposite is true. [99]

Lift Due to Fuselage. The lift of conventional low wing plus fuselage configurations is generally higher by some essentially constant amount than a high wing combina­tion. Depending upon fuselage shape and size the lift – curve slope of the combination tends to be somewhat increased, particularly for mid-wing configurations. An example of such interaction is presented in figure 3. Principle results are as follows:

(a) The lift-curve slope of a wing and fuselage together is some 5% higher than that of the wing alone.

(b) When changing the angle of attach of the wing against the fuselage body (kept at zero angle of attack, the lift-curve slope is some 5% lower than that of the wing alone.

WING 0012 RECTANGULAR A = b/c = 6 FUSELAGE ROUND WITH l/b = 2/3? l/d = 6

a) WING ALONE (NO FUSELAGE ) dC /doc = 0.077

b) WING PITCHED AGAINST FUSELAGE (KEPT AT o< =0) =0.073

c) WING PLUS FUSELAGE TOGETHER = °*080

d) A TAPERED WING (7,c) YIELDS 0.077; 0.074; 0.080

Figure 3. Influence of a fuselage upon the lift of a wing as tested (7,b) in NACA VDTunnell at Rc = 3(10)^ or RF – 8(10)fo.

Regarding (b) it must be mentioned that in the design of airplanes, the wing is usually set at a certain positive angle of a few or several degrees, against the fuselage. It must be noted that such an angle of incidence (denoted by i) produces a certain lift differential (negative in comparison to the lift of the wing alone). However, the lift-curve slope of the wing plus fuselage combination is basically not affected by that angle.

Cross Flow. The fuselage does not fully replace the lift of the portion of the wing “covered”. However, as explained in the Chapter XIX, a cross or “2oc ” flow (5) develops around the sides of every cylindrical or streamline or “round” fuselage body when inclined at an angle of attack against the wind. As a consequence of this cross flow which is increased by the upwash ahead of the wing the angle of attack at, and lift on the wing roots are also increased. However, when rotating the wing against the fuselage operating at zero angle of attack as in figure 3 (test b) the cross flow does not develop and in this case dCL/dai is reduced below the value for the wing alone. In airplanes (where wing and fuselage are rigidly attached to each other) the cross flow effect is so strong that the lift of wing-fuselage combinations grows at first with the d/b ratio, as illustrated in figure 4. After reaching a maximum ratio of ~ 1.06, in the vicinity of d/b = 0.15, the combined lift reduces, however, tending toward zero as the fuselage diameter would ultimately approach the size of the span.

d——————————- 6

Moment Due to Fuselage. As explained in the Chapter XIX the fluid-dynamic moment originating in slender fuselages, consists of two components:

1) From a positive lift on the forebody due to potential flow

2) From the negative lift on the afterbody as a result of the vortex flow.

In conventional wing-fuselage combinations, the angle of attack of the fuselage’s tail within the wing’s field of downwash, is comparatively small. For the purpose of longitudinal stability analysis it will therefore assume that the “second” (stabilizing) component of moment is zero. This is the moment produced by boundary-layer accumu­lation and “viscous” lift originating at the upper (suction) side of the fuselage, near its trailing end. The longitudinal moment of fuselages; in combination with a wing, can then be expected, essentially to be a “free” moment (with no resultant life force). Also from Chapter XIX the destabi­lizing free moment of round fuselage-type bodies in free flow, is approximately

MF /(яМ) = or ;

orCmF=MF/(qCd’)b)= oc° /720 = (*773 (12)

where ’d’ = maximum width, equal to the maximum diameter of the affected volume of air, 5. is the angle of attack in radians S0 = (’d’) тґ/4, and Л – length of the body. Referred to the dimensions of an airplane:

A(dCjdoc) = (S0/S)(i/c)

A(dCWdCL) = (S0/SX^/cXdoc /dCL ) (13)

where (S0/S)(^/c) should not be confused with the tail “volume” of airplane configurations.

a JUNKER, RECTANG LOW WING A = 6, SEE (fc, e)

о NACA, RECTANGULAR MID WING A = 6 (7,b)

• OTHER LARGER ASPECT RATIO COMBINATIONS ■+ JUNKERS, HIGH OR LOW WING, SEE (€>,e)

x DVL, TRIANGULAR WINGS, A = 1.3 & 2, SEE (f>,e)

0 NACA, A = 3 TAPERED, WING COMPONENT (6,g)

AS DERIVED IN (6,e) THE COMPONENTS OF LIFT IN LARGER

ASPECT RATIOS ARE APPROXIMATELY AS FOLLOWS:

a) EXPOSED PORTIONS OF WING: L/Lq = (1 – (d/b))5//J

b) DUE TO CROSS FLOW (UPWASH): L/Lq = (d/b)(l – (d/’b))[100] [101]^3

c) INDUCED ON BODY BY WING: L/Lq = (d/b)(l + (d/b))(1 – (d/b))^3

A SIMILAR SET OF EQUATIONS LEADS TO THE FUNCTION AS PLOTTED

FOR SMALL ASPECT RATIOS.

Figure 4. The lift curve slope of wing-fuselage combinations as a function of the diameter (or width) ratio d/b.

(6) Lift of wing-fuselage combinations:

a) Vandrey, Theoretical Analysis, Lufo 1937 p 347.

b) Multhopp, Fuselages, Lufo 1941 p 52 (NACA TM 1036).

c) See results in references under (7), such as b) & c).

d) Liess, Lift and Downwash Analysis; see (14,a).

e) Hoerner, T Rpt F-TR-1187-ND Wright Patterson AF Base (1948).

f) Lange, Small A’Ratios, ZWB Doct UM 1023/1 to 5.

g) Mayer-Gillis, Load Distribution Among Wing Fuselage Tail, L50J13, L51E14a, L53E08.

h) Analysis is found in NACA RM A51G24, L51J19 & L52J27a.

Wing Interference. The magnitude of the free fuselage moment is increased along the forebody by the upwash induced by the wing. Moment differentials obtained from wing models tested (7) with and without fuselage (essen­tially in mid-wing locations) have been evaluated. Figure 5 shows how those differentials vary as a function of the longitudinal location of the wing. The moment reduces to approximately zero as the wing moves toward the nose of the fuselage and reaches and crosses the theoretical level for a wing position approximately at half the fuselage length. A particular example for tail-fuselage interaction is shown in figure 6. The pitching moment derivatives given are referred to airframe dimensions. The moment ratio 0.0028/0.0068 = 0.41 using theory, equation 12 fits rea­sonably well into figure 5, while the ratio 0.0028/0.0051 = 0.55 from tests of fuselage alone would not. It seems that the body shape alone exhibits a stabilizing “viscous” component A(dCrf/doc)^ — 0.002 (on account of its rather bluff rear end) so that dC^doc = 0.0051 as tested, instead of 0.0068 as predicted by theory.

(+) USING RATIOS LISTED BELOW dCL/da = 0.070

INDICATING FORWARD SHIFT
OF NEUTRAL POINT – дх/с

WING: b/c =5 R = 4(10)5

t/c = 12% RECTANGULAR c

FUSELAGE: 1/d = 7

1/c = 5

d2/S =0.1

Figure 6. Example (17,a) for the contribution of a fuselage body to the pitching moment of an airplane configuration.

Upwash/Downwash. At 1.0, as in the combination

of a horizontal tail surface and fuselage (where most of the body length is ahead of the wing the moment differ­entials plotted in figure 5, surpass the value as in undis­turbed free flow). There is some upwash ahead of the wing, which accounts for this increment of the destabiliz­ing body moment. The downwash behind the wing can, [102]

Fuselage Shape. The longitudinal fuselage edges, such as a rectangular (instead of a round) cross-section shape, in­crease stability slightly when applied aft of the wing and particularly near the tail end and reduce stability when used ahead of the wing. Characteristics of wing-body combinations as affected by the fuselage’s cross-sectional shape, are listed in figure 7. Both the rectangular and the upright elliptical fuselage shape, present pitching moments which are less positive (less unstable) than the round shape. In the elliptical shape, it seems that the forebody causes less cross flow, while the afterbody of the “square” shape seems to be more stable on account of its lateral edges. When adding a radial-type engine in the nose of the fuselage, stability of the combination (without tail) is little affected. All of the configurations shown in figure 7 have fuselage moment ratios below unity. Figure 5 dem­onstrates, however, how much that ratio depends upon the longitudinal position of the wing along the fuselage.

ALL VALUES IN PARENTHESES ARE FOR CONFIGURATIONS WITH WING-ROOT FILLETS, a) AT Rc = 3(10)6 IN VARIABLE DENSITY TUNNEL; Rf = 8(10)6.

Ы WITH 0018/09 TAPERED WING HAVING CHARACTERISTICS SIMILAR TO THE 0012. c) BREAK-AWAY OF LIFT AT CL = 1.0; ALL OTHERS BREAK CLOSE TO CL>

FUSELAGES WITH //b = 2/3 or / = 4 c; ROUND SHAPE AS IN FIGURE 3.

WHEN TESTED ALONE, THE RECTANGULAR FUSELAGE SHAPE PRODUCES A(dC /dCL)= +.03 (BASED ON WING DIMENSIONS, AROUND THE //4 AXIS). THE ROUND FUSELAGE PRODUCES ^(dCn/dCL) = + 0.05.

When applied to the fuselage with rectangular cross – section shape, fillets do not have much of an effect.

Figure 7. Lift and longitudinal characteristics of a wing in com­bination with various fuselage shapes, as tested (7,b, c) m NACA VDT.

The Flying Boat Hull, shown in figure 8 produces, in presence of the wing, positive (bow up) pitching moments which are larger than predicted by equation (12). We may assume that the hard-chined bottom of the forebody (ahead of the CG at ^ 0.4 of the length) may produce some lift, in the same manner as a small aspect ratio wing. Using the corresponding function given in Chapter III the forebody lift, based on the square of the hull beam, corresponds to

dCLb /doc =7/7360 = 0.027

ЛС = – 0.02 ON WING DIMENSIONS = – 0.20 ON BODY DIMENSIONS

Figure 8. Pitching moment characteristics of or due to the hull of a flying boat (7,f). The definition of the coefficient is Cnn = M/(q

bt )•

Fuselage Ducts. The principle stated above, whereby fuse­lage moments are proportional to the volume of air af­fected by them (equation 12) also applies to various other bodies (such as to external stores, for example) and to the addition of such bodies to wing or fuselage. Figure 9 shows as an example, the influence of inlet ducts for jet engines, simulated by adding solid half bodies to the sides of the fuselage. Moments of the configurations referred to the volume parameter (£d2) of the basic fuselage body, are plotted in the graph versus the “span” ratio ’d’/d show,

(a) Lateral ducts with constant length, when exposed to an angle of attack, cause the unstable moment derivative of the configuration, approximately to grow in proportion to the “span” ratio, as postulated by equation (12).

(b) In configurations with the ducts on top of or below the fuselage, the pitching moment essentially remains un­affected. Again, the moment derivative corresponds to the dimension 4d’ measured normal to the direction of angular displacement.

(8) Characteristics of ducts, stores, et cetera:

a) Jaquet, Airplane With Ducts, NACA TN 3481 (1955).

b) Wilson, Twin-Engine Nacelles, NACA W Rpt L-428 (1940).

c) Crigler, Propeller in Pitch & Yaw, NACA TN 2585 (1952); also Comparison with Experiments, W Rpt L-362.

d) Silvers, Influence of Missiles, NACA RM L54D20.

О A = 2 WING 0.6 TAPER RATIO

Stability is the built-in tendency of a vehicle to return by itself to a certain state of equilibrium. For example, a railroad vehicle is extremely stable in regard to pitching (and rolling) motions. Ships have considerable stability (based on buoyancy). By comparison, the stability of all types of aircraft is much more sensitive and complex, resting upon the interaction by means of the flow of air between half a dozen component parts, such as wing, fuselage, propulsion system and tail surfaces in particular.

In this and Chapters XII, XIII and XIV the general charac­teristics of aircraft stability will be discussed. In particu­lar, the stability will be covered in terms of the funda­mental parameters for the longitudinal, directional and lateral modes (1). Included in the treatment will be the basis for determining the stability derivatives of the wing, fuselage, tail and propulsion systems. We will also consider static and dynamic stability of simple and plain aircraft configurations.

(1) It seems that airplanes were flown for years, before their stability was first formulated by Bryan, “Stability in Avia­tion”, London 1911.

1. FUNDAMENTALS OF LONGITUDINAL STABILITY

Longitudinal motions of an airplane, the motions in a vertical plane, can tentatively be analyzed without refer­ence to motions in other directions, on the simple assump­tion that the craft is kept on course and that rolling is prevented by proper control.

Parameters Involved. The aerodynamic forces and mo­ments of an airplane are function of shape, arrangement and interference of the component parts. Their magnitude and possibly their character, may change as a function of speed (Mach number) and altitude (Reynolds number). The resultant stability may also be affected by structural elasticity of the components and/or the configuration. The operational necessity of control by means of the elevator (hinge moments, weight and strain in the system) will also have an effect upon stability. Longitudinal char­acteristics may be altogether different during the landing maneuver (flaps, landing gear, ground effect). Propulsion by jet engines, propellers, or ducted fans cause dynamic interaction problems which have a very considerable in­fluence upon stability. The stability of aircraft motions also involves mass – aerodynamic forces and aeroelasticy arising as a consequence of such motions. Many or some of the parameters mentioned, may or will, therefore, be affected during the motions of which they are the cause and/or integral part. As a consequence, when including all variations of shape, interaction and operation, the number of parameters possibly to be considered, may reach the order of 100. Stability analysis may thus appear to be an unsurmountably difficult problem, particularly in view of the fact that a longitudinal shift of the center of gravity by plus or minus a very few percent of the wing chord, can be the difference between a stable or unstable air­plane. Fortunately, most of those parameters are of sec­ondary importance; many of them have not even been analyzed and/or tested. Some of them are, on the other hand, are of predominant importance such as downwash and possibly thrust in particular. Indeed, consideration of “a dozen” selected parameters will and has to be suffic­ient in the practical analysis of stability.

Notation. As described in the Chapter I, there is a basic convention regarding positive and negative directions of dimensions, forces and moments. The following symbols and conventions will apply. For an airplane as in figure 1, in the longitudinal plane of symmetry, about and in reference to its center of gravity CO there are the geometric factors:

x = longitudinal distance parallel to the zero-lift line of the mean wing section.

z = distance between the wing’s aerodynamic center and the CG, in normal (vertical) direction.

c = mean aerodynamic chord “mac”, replacing the wing aerodynamically.

distance of horizontal tail quarter-chord point from CG.

H = subscript indicating the horizontal tail.

M = CmqSc = pitching moment, usually about the CG.

CP= center of pressure or lift forces.

AC= aerodynamic center of wing (Cm = constant)

There is also a “neutral” point, representing the aerodynamic center of the wing plus tail configuration.

dynamic forces and moments (due to inertia and/or acceleration) may tentatively be disregarded. By neglecting these contributions, the stability of an airplane becomes the favorable interaction between gravity (weight) and aerodynamic (lifting) forces and moments. This type of stability can easily be determined in a wind tunnel by hinging an airplane model about the lateral axis through the point representing the center of gravity. The slope or strength of the restoring moment dM/doi is then called 4‘static stability”. Such investigations provide indispensable information regarding aerodynamic forces and moments contributed by wing, fuselage, propulsion system tail surface etc. and give an insight into the downwash and wake effects. The investigations lead to practical measures of changes necessary for improving stability. Disregarding extreme conditions (such as particularly in more or less “ballistic” missiles) static stability is the most important component of dynamic stability; and as far as airplanes are concerned, static stability can be said to be a prerequisite of dynamic stability.

In particular, a positive pitching moment tends to increase the angle of attack, thus leading to stalling. A positive value of the derivative dCyJdoc or dC^dCu. , means a lack of stability. Therefore, to be stable, every airplane should have a negative pitching moment derivative. The definition of moment arms is more difficult. As far as geometry is concerned, we will consider the distance of any point on the chord of a wings (or on the axis of a fuselage body) aft of the leading edge (or the nose, respectively) to be positive. However, when it comes to pitching moments due to positive lift forces, usually about the CG, the moment arm of the horizontal tail must evidently be counted as negative, while that of the wing (assuming that the CP is ahead of the CG) has a positive value.

Static Stability. The oscillatory motions of an airplane in longitudinal direction (pitching, heaving, variation of speed) can properly be described by a set of equations. Assuming, however, that the motions may be sufficiently slow (thus eliminating all derivatives against time), the

Balance. In airplanes, usually one of the less important parameters contributing to stability is drag. To simplify the consideration of balance and stability, we will thus assume the basic system of forces and moments as in figure 1. Specifically, the lifting or resultant forces are assumed to be normal to the wing section’s zero lift line, through its aerodynamic center AC (which means approxi­mately through the trailing edge). All dimensions are measured in directions parallel or normal to that line. For Cm =0, longitudinal characteristics of the “airframe” are then described by a pair of simple equations, the first of them indicating balance:

L(Ax) + LH zero; L(ax) = L/Th (1)

where the first term represents the pitching moment of the wing and the second term that of the horizontal tail. For forces and positions as illustrated in figure 1, the moment arm ax = g — a, has a positive value, while that of the tail must be counted to be negative. In the form of non-dimensional coefficients (based on wing dimen­sions) balance is defined by

CL (іх/с) = сш (SJS)UJc) (2)

where (SH /S)( J! jc) = vw = horizontal-tail “volume” (be­cause it has the dimension of ft3).

Basic Stability Equation. A second equation describes the variations of forces and moments, as the airframe is slowly rotated (at constant tunnel speed V) about the transverse axis through the CG. For a vertical distance (see figure 1) z = 0, the airplane is then stable, provided that

(dLw /doc)Ли/с I ;> (dl/doc )(Дх/с) (3)

where Ax/c as above, = g — a. Replacing the lift by its coefficient “CL{y and by CLH q(SM/S), stability is indicated by

(dCLW/doc)Vw > (dCL /doc)(Ax/c) (4)

where VM as above. On the basis of these equations, any type of combination of two “wings” might be made “stable”, including tandem and canard configurations. The reasons for preferring one particular type, larger wing plus comparatively small horizontal tail aft for airplanes are:

1) concentration of lift in one large-span wing is most efficient,

2) a “small” horizontal surface is sufficient for stabili­zation,

3) location at the tail promises longitudinal control up to and beyond CLx,

4) location within the propeller slipstream provides improved control at low speeds, such as on the ground.

Equation (4) also means that for longitudinal stability, the sum of the two sides, equal to the derivative of the complete system (for example, with fixed elevator) must be negative; thus:

(dCm/dCL ) = (dCJdocXdoc /dCL ) ^ 0 (5)

Free Moment. Provided that the wing section is cambered (or owing to the deflection of wing flaps) there is usually a free moment indicated by Cm0 = MQ/qSc (independent of lift, and thus also at zero lift). Including this com­ponent, equation (2) reads in non-dimensional form:

Cmg = Cmo+ ((g — a)/c) CL + Cmw = 0 (6)

where g = longitudinal location of the CG, while a = location of the aerodynamic center (including the influ­ence of the fuselage, etc.). The free moment is thus balanced by the horizontal tail. In a conventional airplane configuration, Сгпн is adjusted by suitable deflection of the elevator so that balance is obtained for a particular lift coefficient, corresponding to speed or dynamic pressure when flying. It should be noted, however, that as a con­stant if not changed by slipstream effects, Cm<>does not affect stability. This is the distinction between balance and stability.

Although the equations presented above for the static stability case are simple, determination of the longitudinal stability of an airplane in complex because

(a) various interference effects that cannot be dis­regarded,

(b) the wing downwash has a large effect on the re­sultant lift curve slope of the horizontal tail sur­face,

(c) forces of, and interference by the propeller (or the propelling jet) are needed in the operation of pow­ered airplanes.

Only the simplest type of configuration without wing flaps and without propulsion effects will be considered in the following sections of this chapter. The influence of the propulsion system on longitudinal stability is covered in Chapter XII.

DYNAMIC LONGITUDINAL STABILITY. After an air­craft is disturbed from its flight path two basic motions are obtained: 1) a long-period or phugoid oscillation and 2) a short-period oscillation. These oscillations may be developed even though the static longitudinal stability of the aircraft is satisfactory. The long-period phugoid oscil­lation may involve several seconds depending on the par­ticular airplane. This motion is normally hardly noticed by the experienced pilot who automatically damps it out with the stick or power changes. The short-period oscil­lation, however, must be adequately damped in the basic airplane system for satisfactory dynamic longitudinal sta­bility. If the airplane has an undamped short period oscil­lation, attempts to correct the difficulty can lead to pilot induced oscillations, PIOS, which accent the unstable con­dition.

The analysis of the dynamic stability problem requires the use of the basic equations of motion and the identi­fication of many stability derivatives. With the stability derivatives involving velocity, angle of attack, q, elevator angle, compressibility, moments, etc. the dynamics of the airplane are analyzed for at least the stick fixed and stick free cases. The development of the basic equations of motion and their analysis in terms of dynamic stability are covered in many available sources, for instance (2,fog) and, therefore, will not be repeated here. Rather a basic understanding of the static longitudinal problem is con­sidered. From the material presented, the derivatives need­ed for the dynamic analysis can be determined with a greater degree of accuracy.

After dealing with all the details of control effectiveness and hinge moments, there is one influence left to be discussed, namely compressibility and/or Mach number.

Prandtl Glauert Factor. The lift-curve slope of foil sec­tions (not that of wings having finite aspect ratios) grows in proportion to the Prandtl Glauert factor

PG = 1/11 – M*~ (25)

Experimental evidence as in (22) proves that due-to-flap lift-curve slopes dCL /doc, also tend to increase in propor­tion to PG, although the boundary layer can have an influence reducing this effect.

Hinge Moments. Usually, hinge moments due to flap de­flection are proportional to dCL /dcf. Hinge moments may, therefore, be expected also to grow in proportion to or possibly at a lesser rate than PG.

Spoilers. The effectiveness of suction-side spoiler devices usually increases with the lift coefficient. As a conse­quence, effectiveness may be expected to grow in propor­tion to PG. Experimental results listed in (9,p) confirm the prediction, although tunnel blocking (choking) seems to exaggerate the influence.

Critical Mach Number. The Prandtl-Glauert rule cited above ceases to apply as soon as at any point of the foil-and-flap section the velocity of sound is first reached. The maximum local velocities are a function of thickness ratio, lift coefficient and section shape. Considering a deflected and lifting control flap, a critical point is found at the suction side of the flap near its nose where the contour makes a bend. Particularly critical conditions (high local velocities) must be expected at the exposed side of an overhanging nose balance. The balancing quali­ties of such noses may thus be limited by compressibility, and the flap effectiveness may be expected to be reduced at the same time.

(22) Influence of compressibility upon flap characteristics:

(a) Stevenson, 9% Foil Sections, NACA TN 1406 & 1417 (1947).

(b) Hammond, Flaps and Spoilers, NACA RM L53D29a; also RM L56F20.

(c) Lindsey, 9% Foil with 30% Flap, NACA RM L56L11.

(d) Lowry, Rectangular Wings, NACA RM L56E18.

(e) Whitcomb, Transonic on Swept Wing, NASA TN D-620 (1961).

(f) MacLeod, Bump Tests, NACA RM L50G03.

(g) Using free-flight rockets, NACA RM L48A07 & L48I23.

(23) In a low-speed water tunnel, the time scale is considerably larger than in a wind tunnel (at one and the same Reynolds number).