Typical streamline bodies are unstable. Devices such as missiles, bombs, airships and fuselages can be stabilized by adding fins to their rear end.

Static Stability. As stated in context with equation 22 there is a range of the angle of attack (or yaw) around zero, where lift or lateral force and moments vary in linear proportion, so that dC^/dC*. = constant. This derivative represents the minimum of the static stability (33). When using a moment coefficient referred to the bodies length (subscript^) the neutral point (where the moment is zero, and/or about which the static stability parameter may be constant) is located at

Д X/-C = dCw/dCL

Подпись: neutral point Подпись: ОПодпись:Подпись: np /Подпись: - o.0124 ,STABILITY OF STREAMLINE BODIESПодпись: ' o.020 Подпись:where A x = distance measured from the center of gravity or the reference point about which the moment is tested, positive when downstream. Since stability of bare stream­line bodies is usually “negative”, the neutral point is forward accordingly. In fact, it may be located far ahead of the body’s nose, such as x = —^, for example.

The Cones as in figure 1, have a fairly constant center of pressure, equal to neutral point (or aerodynamic center), located between 64 and 65% of their length. Cones thus behave in the same manner as slender delta wings. Placing the CG slightly ahead of 64% of the length, conical bodies could thus provide stable lifting “vehicles”. The circular type in figure 24 has a maximum L/D = 2.4, at Cl 0 = 0.4, id est based on base area. The L/D ratio could be im­proved by giving the cone a flat (elliptical) cross-section shape, of the type as included in figure 2, for example.


Figure 30. Stability (position of neutral point) of the essentially cylindrical “scout” rocket vehicle (38,a) for three different condi­tions of the tail end.

Cylindrical bodies with some type of “streamline” nose, could be expected to have their neutral point within the length of the nose; roughly as follows:

at 2/3 of nose length, for conical shape at 1/2 of length, for ogive (as in figure 15)

Because of the “viscous” type of lift developing in the cylindrical afterbody (see equation 8) the neutral point may be further aft, however. For example in the shape at the bottom of figure 14, the point is at or slightly behind the juncture between nose and cylinder.

A Flared Tail End as in the second configuration in figure 30, is theoretically expected to provide a stabilizing force the magnitude of which corresponds to equations 1 and 3,

dCLd /doc = (SB /Sd -1) 0.0274 (26)

where “B” indicates the base area. The tested differential ACNo = 0.02 agrees with this equation sufficiently well. Moment and neutral point can be calculated as they were tested.

= 6.7 d

Cn = "^"/qd2^ about o.5 point

Су = Y/qdC p = angle of aideslip

Figure 31. Longitudinal characteristics of three streamline bodies of revolution (9,h) differing in the shape of fore-and afterbody.

Boat Tail. As soon as the afterbody is made tapering, it begins to produce a negative lift force, thus making the moment more unstable (tail down, nose up) and causing the neutral point to move forward. Figure 15 is an ex­ample, where the addition of a boat tail results in a neutral point location ahead of the body’s nose. The shape with the cylindrical afterbody in figure 26, will be stable if placing the CG ahead of x = 0.2. Boat-tailing (reducing the width of the afterbody) reduces stability. In fact, stability of the second shape in the illustration (still having a blunt base) is almost as poor as that of any pointed tail shape (such as an airship hull, for example). This result is confirmed in figure 31 where characteristics of three same-size bodies are listed, differing in shape. The blunt-based body has nearly the same derivatives as the slender double-pointed type.

Streamline Bodies. Although the results in figure 31 may quantitatively be deficient (23) they nevertheless demon­strate how stability deteriorates when “streamlining” the shape:

When reducing the afterbody to a point, the neutral

point moves forward.

when giving the forebody a fuller shape, the point moves still further forward.

As a consequence, “perfect” streamline shapes such as in figures 32, 33, 34 have neutral points up to about once their length ahead of the nose.

Thickness Ratio. Characteristics of two streamline bodies differing in thickness ratio, are presented in figure 32. While, this time based on area (b.|), the lift-curve slopes are roughly the same, the fatter body is much more unstable than the thin one. Neutral points of various types of bodies are plotted in figure 35, evaluated from tested pitch and/or yaw derivatives show:

a) round streamline bodies with a tapering afterbody ending in a point are most unstable.

b) airship hulls (bare, without fins, but with polygonal edges) still have a position of the neutral point more than half of their length ahead.

c) sufficiently blunt-based “fuselage” shapes are the least unstable; their neutral point may be at their nose point.

The mechanism of negative lift in the afterbody (see figure 1) evidently deteriorates as the boundary layer grows with the length ratio ijd. At d/Л —0, we may speculate that the center of whatever linear lift com­ponent might still exist, is somewhere within the nose part.

STABILITY OF STREAMLINE BODIESПодпись: Figure 33. Longitudinal moments and neutral points of streamline body of rotation (39,c) with and without tail fins attached, tested in water towing tank. d = o.10j[


STABILITY OF STREAMLINE BODIESПодпись: Figure 32. Longitudinal characteristics of a pair of streamline bodies (9 j) differing in fineness ratio.

Подпись: Neutral Points of various types of bodies are marked in figures 26, 31, 32 and 36; and they are indicated in figure 27 by limiting horizontal lines around zero angle of attack. The distance from the nose point, is on the average 23. X 3 II і о for “perfect” streamline bodies = -0.5 | in turbulent air, or with surface roughness = -0.31 for blunt-based parabolic-arc bodies = -0.1 і! for boat-tailed cylindrical shapes = +0.2 1 for cylindrical afterbody shapes It seems that laminar boundary layer flow (around the

nose) results in more unstable characteristics, while sur­face roughness is stabilizing. Stability about a given point such as the center of gravity, corresponds to

dC^ /dC. L =(хг1/^-(х9Д) (27)

where Xg = downstream of the CG from the nose of the body. A body is stable when the derivative is positive, id est when the CG is ahead of the neutral point.

Wedge Shape. Without fins, streamline bodies are hope­lessly unstable. For example, the derivative of the bare and smooth airship hull in figure 37, about the CG at x = 0.45.|, is dC^jldC^ = -1.55, which means that the neutral point is 1.1 body lengths ahead of the bow. From efforts to improve the hull shape, the configuration as in figure 38 was investigated and indicated:

a) When giving the tail end a wedge shape, with a vertical trailing edge, both the basic linear and the “second” non-linear lateral force components are increased..

b) Analysis confirms that the additional forces originate in the rear end, approximately at a “center” as indicated in the drawing.

c) As a consequence of (a) and (b) bodies with a flat tail end are less unstable. When making the trailing edge, h = d, the hull might obtain a stability similar to that of ogive-cylinder combinations.

Подпись:The yaw-moment derivative of the airship hull and those of a few other “fuselage” bodies having a wedge-like afterbody shape, are plotted in figure 39. The results correlate with equation 3. A modification similar to that in figure 37, is shown in figure 38 of Chapter XIII. In that case, the neutral point is located ahead of the body’s nose, at

хД( =-1.2 for plain body of rotation








______ М2 сьь



= 245 m

V = 80 MPH V = 0.5 b2

J = 6 d Rj = 5 (10)”






Cx = °-07

Подпись: 0.04Подпись: cmbf 2Подпись: I ABOUT CENTER OF BUOYANCY AT 0.45 Подпись:STABILITY OF STREAMLINE BODIES

Подпись: dCmT /dee = (dCLT I doc Хіт ЦХ Sr I50)

Figure 37. Longitudinal stability characteristics, with and without fins, of LZ-128 “Graf Zeppelin” airship (34,a).

Dorsal Fins are usually meant to be comparatively narrow surfaces attached to the tail end (or possibly the front end) of fuselages. Forces and moments of a plain “fuse­lage” model are shown in figure 40, without and with such fins (strips of metal) attached to the afterbody. With these strips there is a modest influence upon lift and moment near zero angle of attack. Also the non-linear lift or normal-force component is considerably increased along with improved stability at the higher angles of attack.

In conclusion, “dorsal” strips are primarily effective at higher angles of attack. The normal-force increment is roughly Д Cw — sin2 oc. In figure 40, the cross flow co­efficient of the part of the afterbody “covered” by the fins (on projected lateral area SF ) is in the order of

Cc =(dCM/d(sin2cK))(b^/SF)=3

Tail Fins. A body with the round and full forebody such as shown in figure 30 is the most unstable kind. There is evidently no “support” in the afterbody; and the fore­body force is located comparatively far ahead. The only aerodynamic means of stabilizing such streamline shapes, are fins attached to their tail end. The longitudinal (pitch or yaw) moment of a streamline body (similar to the hull of an airship or a submarine) is plotted in figure 33. Static stability about the center of buoyancy or gravity, is only obtained after adding the largest fin shown (number IV). The stabilizing moment of the tail fins (subscript “T”) corresponds to

The lift curve slope might be predicted for two extreme cases:

a) For small aspect ratios, equation 3 suggests a dCfb /doC = 0.0274. The contribution of tail number IV is some 10% larger than this, possibly because of favorable 2-alpha type body interference.

b) In the range of larger fin aspect ratios, it can be assumed that the effective ratio is Vl the geometrical one (36). Using equations of Chapter III, the lift-curve slope can thus be obtained. The contribution of tail number I (with A = 4.4) agrees well with this type of estimate.

Fin Size. When adding fins to a streamline body, the neutral point moves aft, say from хЦ = -1 (ahead of the nose) possibly to хЦ = +0.5 (half way between nose and tail point). The difference Д x/f = 1.5 is the length stabi­lized by means of the fins. In figure 33, it is seen how stability (position of the neutral point) varies as the chord length of the fins is increased. The configurations shown in figure 34 demonstrate how stability increases with the span of the fins. It may be speculated that the stabilizing forces originating in the fins grow in proportion to their chord length and to the square of their span. Neutral – point positions were, therefore, plotted in figure 41 as a function of the tail-size parameter (b/d) (сЦ). The func­tions thus obtained can empirically be interpolated by

А хЦ = (dC^ /dCL ) = 2.5 V(b/d)2 (с/Д)" (28)


Figure 39. Lateral force derivatives of streamline bodies, as a function of the height of the trailing edge (if any).

(32) ARC, Characteristics of airships, pitching and turning:

a) Pannell, R-32 Turning, RM 668 & 812 (1921).

b) Simons, Stability and Turning, RM 713 (1919).

c) Jones, Stability, RM 751 & 799 (1922).

d) Jones, Equilibrium in Turns, RM 716,749,781,78 2 (1922).

Ring Surface. Lift and moment (about the CG as in­dicated) of a streamline fuel tank (to be dropped when necessary) are presented in figure 34. The “small” set of fins is evidently not sufficient to stablize the body. Neu­tral stability is only obtained after adding a small “ring” to the larger set of fins. The model of a 1000 kg bomb was tested with two different tail surface configurations. As shown in figure 42, addition of a ring around the four fins, increases stability. The neutral point is moved aft on account of the ring by Л x = 0.16_f. It can be assumed that the increment of the stabilizing forces in the tail surfaces are obtained on three counts:

1) the ring serves as an end plate, thus increasing the effectiveness of the fins;

2) the ring surface develops stabilizing forces of its own (see equation 20);

3) the ring reaches out into the flow outside the body’s boundary layer (wake).

In the configurations as in figures 34 and 42, the stabi­lized length is increased by the addition of a ring, roughly by 10 and 20%, respectively. This small change can make the difference between stable and unstable configuration.

Airship. Stability characteristics of an airship are shown in figure 37. Without fins, the slope near zero angle of attack (or sideslip) dC^ /dCL = —1.4 means that the neutral point is roughly one length ahead of the nose. After adding the set of fins, that point is brought back as indicated, roughly half way between nose and the center of buoyancy The airship is not stable in this condition. The speed of these vehicles is evidently low in comparison to their mass; which means that their Froude number is comparatively low. As a consequence, the deviation from their direction of motion takes place at such a slow rate that correction by manual or automatic control (rudder and/or elevator) deflection) can easily be accomplished. As shown in 30,f one particular airship needs alternating rudder deflections of plus/minus 4° to keep it on course. Although it would be possible to make the airship in figure 37, stable by increasing the size of the fins, it was evidently concluded that this would not “pay”. Larger fms would add to drag and weight. As we will see later, the maneuverability (turning) would also be reduced when increasing stability.

(33) For a discussion of the dynamic stability of airships see (30,f) and (32,e). for example.

(34) Investigations in AVA wind tunnel:

a) Kohler, “Zeppelin” Airships, Ringbuch D’Lufo Rpt IA15 (1940).

b) Kohler, Airplane Skis, Luftwissen 1937 p 6.

Tail Fins. When attaching fins to the stern of the airship hull as in figure 21, they first produce some circulation – type lift. The linear normal-force slope is doubled; dСд, ь /doc = 0.003. Based on various areas, the increment (due to fins) has the derivative:

A( dCN/doc) =0.003

on lateral hull area (d )

= 0.024

on hull di­ameter area d

= 0.055

on exposed fin area

= 0.026

on fin plus

“ с о vered”


The most reasonable definition is the last one, on “total” fin area, including the part of the hull between them. The slope somehow corresponds to the geometric aspect ratio Ар = 0.7. There are interference effects involved, how­ever:

a) At the sides of the hull, the angle of attack of the fins is possibly doubled, on account of “2 alpha” cross flow.

b) As explained for horizontal tail surfaces, in Chapter XI, the boundary layer (wake) developing along the hull inter­feres with the circulation around the fin plus stern con­figuration.

The stabilizing moment due to the lifting forces in the stern, corresponds to

dC^/doC – (dC^ /dec )C/T/T) (29)

where ” moment arm of the “tail” to the axis of reference. Solving the results as in figure 21 for that arm, it is found that the location of the stabilizing tail force is at

x/* = (dC^/doOM (dCN)/doc) =
0.0013/0.0015 = 0.89

which is as expected, somewhat forward within the area of the, and covered by the fins.

(35) Dorsal and other fins on fuselages:

a) see Bates in reference (10,b).

b) Hoggard, Around Plain Body, TN 785 (1940).

c) Recant, Fighter Model, NACA W’Rpt L-779 (1943).

Stability. The center of gravity of an airship is or has to be identical with that of the buoyancy. In the example as in figure 21, the center is at 46% of the length. The stabi­lizing effect of the fins at the end of the hull corresponds to a moment arm Jp Ц= 0.47. When shifting the axis of reference to the nose point, dC^/d/} is just about zero. For practical purposes, this means that the airship would be stable if it would pitch about the axis through the nose point. Stability (33) improves, however, when reaching higher angles of a ttack, where the fins provide a “second” and non-linear normal-force component. Extrapolating corresponding to equation 14 the ACN due to fins as in

The last definition yields a very reasonable value for the fin configuration considered.


Every slender streamline body has theoretically an un­stable “free” moment. Because of viscosity and the flow pattern of the type as found with slender wings, the moment is reduced, however. Therefore, in this section we will discuss the longitudinal moments including the neu­tral point of various types of streamline bodies. It should also be noted that in an isolated body of revolution, forces and moments due to an angle of sideslip 2?, are of the same magnitude as those due to a same-size angle of attack.

Theory. The load distribution of a slender streamline body is shown in figure 3. There are positive lifting or normal forces in the forebody and as predicted by theory, forces equal in magnitude, but directed downward (nega­tive) in the afterbody. The corresponding free moment (no resultant force) is

М = -0.5р V2 kV sin(2oc) (15)

where к = 1, for fineness ratios-f/d above 20. The minus sign indicates in this text, that the body is unstable, that the moment tends to turn the body away from the longi – tudianl direction of the flow. “V” indicates the effective volume of the air clinging to the body, so to speak, in lateral motion. In case of a body of revolution, this volume happens to be equal to that of the solid itself. Within the range of small angles of pitch or yaw, equation 15 thus reduces to 44:

M = -2 к q V (У ; and to

0^= M/qV = —2k otradians (16)

Streamline Bodies. As indicated by theory, the free mo­ment of streamline bodies reduces with their fineness ratio. The corresponding factor in equation 16 may be approximated by

k = (l-b/0 (17)

For ЪЦ = &Ц = 1 (suggesting the shape of a sphere) the moment thus reduces to zero.

Pressure Distribution. The pressure distribution of the ellipsoidal body in figure 22 is shown as tested along the meridian, both at the bottom (at 0°) and on the top side (at 180 ). It should be noted that the distribution as predicted by theory is symmetrical. In other words, if reversing the direction of flow, the pressure distribution of this or any other streamline body would theoretically be the same. The tested distribution agrees fairly well with theory, along the forebody. Approximately at 3A of the body length, the pressure reduction, suction, expected by theory to take place on the lower side, begins to be deficient. This means that negative circulation around the tail end is reduced. At the upper side, divergence starts roughly one diameter ahead of the maximum thickness. All along this side of the afterbody, the theoretical in­crease of the pressure toward a rear stagnation point, fails grossly, although both lower and upper side reach to­gether a positive pressure level in the order of Cp = +0.1. The area enclosed between the distributions on upper and lower side is an indication for lift or normal forces. While in the forebody, the force is up, essentially as described by equation 3, the normal force in the afterbody is almost zero (at oC = 20° as tested). The consequence is a re­sultant lifting force, and a reduction of the unstable moment (about the point at 0.5 of the length) to roughly half the theoretical value.


Подпись: (19) Slender bodies at high angles of attacks: a) Letko, Lateral Characteristics, NACA TN 2911 (1953). b) Polhamus, Directional Stability, TN 3896 (1956). c) Maltby, Smoke Studies, RAE TN Aero-2482 (1956). (20) Model investigations of bare airship hulls: a) Jones, New Shape, R-33 Model, ARC RM 802 (1922). b) Jones, Model R-38 Airship, ARC RM 799 (1923). c) Abbott, Models in VDT, NACA T Rpt 394 (1931 к d) Jones, R-101 Airship Model, ARC RM 1168 (1929).


a – 0.5 rt


crap “ °-018 ” °-020 – 0.056

cnae – – о-009 – -0-007 ■ -0-020

сї4 – r/,a2 cnb – – HVqd^ j(-7.5d

Я.-сХ A— і 0Л7 Ji

Figure 23. Directional characteristics of a wing-fuselage configura­tion as affected by a pair of “ducts” (12,a).

Elliptical Cross Section. The volume of air affected by a streamline body when at an angle of attack or sideslip, corresponds to its thickness measured normal to the plane of angular displacement (to the “span” when at an angle of attack). For example, a fuselage body with elliptical cross section will exhibit different moments depending on whether the body is placed at an angle of attack or of sideslip. Tentatively, lift and moment will be proportional to the square of the respective “span”. Therefore, and since the volume may not always readily be known, the most appropriate moment coefficient will be:

C^M/qbV (18)

where b is meant to be the “span” or “beam” when at an angle of pitch, or the height or thickness of the body when at an angle of yaw. As stated in the beginning of this chapter, in conventional streamline bodies, the so-called block coefficient may roughly be:

“B” = У/ЦЪ h) = between 0.5 and 0.6

where b = breadth or beam and h = height. In bodies of revolution h = b = d. Referring the moment above, to the nominal volume (b2^) of the body, we obtain the co­efficient

(W= M/qb2^ =

– (2^/180) к “B”o£ = -0.035 к “В” od (19)

(22) Aerodynamic characteristics of seaplane floats:

a) Liddell, Edo Floats, NACA W Rpt L-722 (1942).

b) Parkinson, Hydro – and Aerodynamic, NACA TN 656 & 716 (1938).

(23) For some reason, the neutral points in Figure 22 are less forward than those in figures 27 and 28.

EXPERIMENTAL RESULTS. In streamline bodies whose displacement volume is directly “useful” such as in air­ships, the center of buoyancy, tentatively identical with the center of gravity, is a convenient reference point for all moments. Moment derivatives of bare airship hulls, as tested about this point, are presented in Figure 13. The tested values are between 70 and 75% of the theoretical function using equation 19 with “B” = 6. In typical streamline bodies (with volume concentrated more for­ward) the theoretical pressure distribution (l, h) changes from “positive” to “negative” at the position of the maximum thickness. Thus, the corresponding point on the body axis, is the natural reference point. Experimental results of such bodies are also given in figure 13. Within the range of higher diameter/length ratios, agreement with equation 19 is obtained when using “B” between 0.30 and 0.35, instead of the geometrical block coefficient for such bodies (which is in the order of 0.5). In other words, the moments are between 0.6 and 0.7 of the theoretical function. However, at diameter/length ratios approaching zero, higher fineness ratios, experimental moments are smaller. It appears that the boundary layer along the afterbody grows with the body length, thus reducing the negative normal forces developing along the afterbody; see figures 3 and 22. In fact, we might speculate that the moment may only be half the theoretical at d// ^ zero.

“Ducts”. When adding external stores, canopies or similar, essentially “streamlined” volumes to the outside of a fuselage, the changes in normal force and moment due to angle of attack or sideslip, can be predicted or estimated using the principles and equations presented above. A pair of “ducts” (simulating the presence of jet engines, but without openings and without any flow through them) was added to a plain fuselage body (12,a). The pitch moments and the yaw moments of the configuration are presented in Chapters XI and XIII. In sideslipping motion (where the wings may be assumed not to interfere very much), the tabulation in figure 23 demonstrates:

a) With the ducts at the sides of the fuselage, the yaw derivatives remain basically unchanged. Small differentials are due to viscous interference.

b) When placing the ducts on top and below the fuselage, lateral forces and yaw moments are considerably in­creased.

On the basis of equation 3 and test data the lateral force increases in proportion to (h/d)2 = 1.75?=^ 3. The value of the negative, unstable, yaw moment is increased to a lesser degree, evididently because the force-producing vol­ume of the ducts is concentrated near the CG of the configuration.

Подпись:Ducted Body (15). We do not have suitable experimental results at hand, at this time, showing the influence of an axial duct upon lift and longitudinal moment of a slender body. Considering an open cylinder with unobstructed axial flow through its length, the deflection of the volume of air produces theoretically

CLo = 2tf = (‘Гґ11 SO) ot° = 0.035 ot° (20)

where the coefficient is based upon frontal area. Assuming that the turning takes place at or near the rim of the inlet, the pitch moment corresponds to the arm to the point of reference. If this point is a x = 0.5^, the maximum of the increment corresponds to the coefficient (based on body lengthy)

Сщ = – a: = -0.0175 a:° (21)

Considering a streamline body with a duct smaller than cross section, and with a-velocity through the duct lower than the ambient spped, the values above have to be reduced in proportion to velocity and area ratio. The lift differential can be appreciable (100%); the moment due to duct flow is destabilizing.



The ring wing shown in figure 24 can be considered to be a ducted slender body:

a) The unobstructed duct is expected to provide a lift component corresponding to equation 28. Based upon nominal duct area d equation 3 is obtained, dCLb/dod =0.0274.

(29) ARC, aerodynamic load distribution on airships:

a) Jones, Pressure Distribution R-33 Model, RM 801 (1922).

b) Bairstow, Aerodynamic Bending R-38, RM 794 & J’RAS July 1921.

c) Distribution on fins and rudders, RM 779,808,911 (1921).

d) Jones, Aerodynamic Loading in Turns, RM 780 (1922).

e) As reported in ARC RM 775, the airship R-38 (Zeppelin Reparation “2”) broke in two, while turning at a speed of some 50 kts. As deduced from experimen ts in (a) & (b ) lateral forces are concentrated in the bow (equation 3) and in the stem (due to fins and rudders).

b) The outside shape is that of a streamline body with cut-off rear end boat tail. Theory suggests that lifting forces on that outside correspond to equation 3.

c) The combined lift coefficient (based on the square outlet or base diameter) is theoretically

dCLb /dcx: = 2(0.0274) = 0.055 (22)

The tested value is practically the same.

Assuming that the lift derivatives of ring wings with other aspect ratios A = d/c or d/T, also tested in (21,c), also be twice that of solid bodies (having same base area) the half values of the resulting derivatives have been plotted in figure 6. Agreement with other results in that graph is excellent if doubling the fineness ratio Ц/Ъ).




7 NACA T/d = 6.8 AIRSHIP HULL (20,c)








Подпись: (C) OGIVE PLUS CYLINDER BODIES 0 TN 3715, J/d = 13, AT M = 2 (14,a) • NASA TN D-1297, J/d = 11 AT M = 0.4 (14,c) Figure 25. Longitudinal trim points (where moment = zero) of various types of streamline bodies, as a function of their angle of attack. V NACA "PARABOLIC" BODY WITH BOATTAIL (17,e)

Neutral Point. The ideal moment of streamline bodies (without any resultant force) is “free” (independent of the point of reference). Since in real, viscous fluid flow, every body does develop lift (or a lateral force) when at an angle of attack (or of yaw) the longitudinal moment depends upon the axis of reference. To say it in different words, the resultant force produces a moment about the center of gravity or whatever axis is used. The location of the resultant force can only be obtained from pressure distribution tests, as for example in figure 3. Usually, therefore, its moment contribution is not known.. Another method of describing the combined moment, is to quote the point on the axis of the body, for which the combined moment is zero. For small angles of attack (where both lift and moment vary in linear proportion) the moment reduces to zero at the neutral point, the location of which is away from the axis about which the moment was tested, by

A xty = dCM/dCL = (dCyry/dot )/(dCL /da;) (23)

For example, the neutral point of the bare airship hull in figure 21, is approximately at x = —ahead of the body’s nose.


Zero Moment. The location x for the zero moment is determined by

A x/f C^/Q. (24)

In such points, the body is in trimmed condition, which does not mean that it would be stable. The point for zero moment changes in the manner as shown in figure 25. The most forward location of every particular body is around zero angle of attack, as defined by equation 23. In “typi­cal” streamline bodies (with a tail tapering to a point) the neutral point is far ahead of the nose. As the angle of attack is increased, the point moves back, it reaches the body nose at angles around 20 . Depending upon the location of maximum thickness, the point may then ap­proach a position between lA and Й the length of the body.

Cylindrical Streamline Bodies. An ogival or conical streamline body can be expected to produce the circula­tion-type linear lift component. This lift force acts within the length of the nose shape. Characteristics of one such body are presented in figure 26. As the second, non-linear normal-force component develops, the center of the com­bined force will move downstream. If the linear lift com­ponent is located at 0.4 of the nose length, and the cross-flow component assumed to be centered at lA the length of the cylindrical afterbody, the combined center of pressure is found to be as tested, for example near 50% of the length at angles of attack between 16 and 20 .

Подпись:Nose Shape. Results of a cylindrical body are presented in figure 14, for several shapes of the forebody:

a) As soon as a small rounding radius is provided (as small as 10% of the diameter) the drag coefficient CDo drops from 0.9 to less than 0.2.

Comparison of a pair of streamline bodies (18,a) as to their stability derivative Ax/( =(dCmb2^/doc)/(dCLb/doO, taken within the range of Ы = +— 8°, or CLb = +- o.15.

Figure 26. Influence of afterbody shape on the longitudinal (or directional) characteristics of a fuselage body.

(30) Aerodynamic characteristics of airships:

a) Klemperer, Stalling, J Aeron Sci 1934 p 113.

b) Freeman, “Akron” Pressure Distribution, NACA T Rpt 443.

c) Frazer, “Full Scale Experiments”, J. RAS Sept 1925.

d) References appended to Jones, J RAS Feb 1924.

e) Rizzo, Static Stability, NACA TN 204 (1924).

f) Arnstein and Klemperer, Aerodynamics and Stability of Airships, see Durand’s “Aerodynamic Theory”, Volume VI.

b) The normal-force derivative is roughly independent of nose shape. The experimental values are slightly higher than the dC^ /doc = 0.0274 as indicated by equation 3; see equation 8.

c) Disregarding the really blunt shape, the pitch moment derivative is approximately independent of nose shape. On the average, the aerodynamic center is at

A x/j = dC^/dC^ = 0.012/0.030 = 0.4

ahead of the reference point. In other words, the center of lift within the range of small angles of attack (say to plus/minus 4°) is at x/.( = 0.57 -0.40 = 0.17, or at x/d = 1.6, behind the nose point.

Afterbody. The experimental results in figure 15 give a perfect illustration of slender body aerodynamics:

a) the ogival forebody produces the theoretically expected lateral force (equation 3).

b) a cylindrical afterbody may produce a small com­ponent of force, of the type as described in context with equation 8.

c) a boat-tailed afterbody produces a negative component of lift, in the manner if not magnitude, as predicted by theory.

Curvature. An airship or a submarine when turning, moves into a curved stream. As illustrated in figure 28, geometric conditions may then be such that nose and forebody are at an angle of yaw close to zero, while at the stern and the tail fins, the angle is in the order of 37°. This condition can roughly be simulated by testing a curved or cambered body in a straight stream of air or water. There are also bodies which are cambered naturally, such as the float in figure 25, or certain fuselages.

Подпись: Figure 27. Aerodynamic characteristics of an airplane float (22,a). Cylindrical Middle Body. There are streamline shapes such as possibly in airship hulls or fuselages, where a con­siderable part of the length is cylindrical. Basic theory expects destabilizing lifting or normal forces to originate in the bow and the stern, while the cylindrical middle body is not supposed to contribute to force or moment as far as small angles against the wind are concerned. Since the bow and stern have a considerable moment (about the center of buoyance or the CG) such partly cylindrical bodies may have larger moments than truly streamline shapes (with the same fineness ratio).

Cross Section. A series of four “fuselages” differing in the shape of their cross section are reported in 14,a. As shown in Figure 12 of Chapter XI, the lift and pitch moment essentially a function of the body width. A fuselage with a flat and wide shape has considerably larger pitch mo­ments, accordingly, while a “deep” and naroow body has reduced moments, defined about the point at x/^ = 0.54. As series of 4 blunt-based bodies is presented in figure 12, differing in the span/thickness ratio of their cross sections. The pitch moment about the point at x = 0.58^ , properly referred to b2 and^ , reduces from 0.007 to 0.003, as the span ratio is increased to 3. The neutral point moves downstream from хЦ = —0.8 (ahead of the body’s nose, for the body of revolution) to хЦ – +0.2 for b/t = 3.

Seaplane Float. Not all streamline bodies are rotationally or otherwise symmetrical. Aerodynamic characteristics of a float are presented in figure 27. Lift and moment seem to vary in the same manner as in other bodies. There is an irregularity, however, in the moment function, within the range of negative angles of attack. It is suggested that the flow separates here from the bottom of the float, behind the step.

Camber. Characteristics of a cambered body are presented in figure 29. For the linear circulation type of lift, the zero angle of attack corresponds to the tangent at about 3A of the forebody length (near the point where the fore­body lift may be centered). The slope is the same as that for the straight body. For the non-linear lift component, the average angle of attack of the afterbody can be ex­pected to be responsible. This angle is (a: + A oc) and the non-linear normal-force component may be

(ot + A oC)Z

THE LONGITUDINAL MOMENTS OF STREAMLINE BODIESПодпись: Figure 29. Longitudinal characteristics of a “cambered” streamline body (39,c) obtained from the body of revolution in figure 33, by curving the mean line, or center line. Подпись:In the body as in figure 29, camber is “с”Ц&&0.5 djj( = 0.5/5.8 = 0.086. This ratio can be understood to be an angle equal to 5°; id est 0.086 (180/^гґ) « 5°. The non-linear lift component as tested, can be explained by

Z CL (o’ -3° + 4°)Z

where 3° = zero-lift angle of attack, and 4° = similar to the 5 corresponding to camber. For example, at o~ = 19 —3 +4 = 20 (as in part C of the graph), the non-linear component is approximately the same as at oC = —19 —3 +4 = —18° (as plotted in the form of +18° in part A of the graph).


Figure 28. Geometry and lateral forces of an airship (or of a submarine) during a turn. Dimensions are assumed in such a manner that the stream curvature corresponds to the camber of the body in figure 29.

(31) Characteristics of airship fins:

a) ARC, Model Results, RM 714, 799, 802, 1168.

b) NACA, Technical Reports 215, 394, 432.

c) Pressure distributions, ARC RM 808, 811, 1169.

d) Pressure distributions, NACA T Rpts 213, 324, 443.


Streamline bodies and similar elongated shapes are used wherever the drag must be kept to a minimum. These bodies are, therefore, used for all types of missiles, aircraft fuselages, engine nacelles, airships, fairings and any other object that should have minimum drag. In addition to the drag force streamline bodies produce a lift force and longitudinal or yaw moments which are covered in this chapter.

Geometry. When considering aerodynamic characteristics of streamline bodies as such, their coefficients are to be based on the dimensions:

d = diameter (if circular as in a body of revolution)

b = width (span) or height, normal to angle against flow

Л = overall length of body

x = distance from nose, along axis.

The most important dimensions are b and JL; and we will preferably use the coefficients


CNb = N/qb for lift or normal force

Cm^= M/q ib for longitudinal moment

It should be noted that in an isolated body of revolution, there is no difference between the derivatives of pitch moment (dC^do/) and of yaw moment (dC^/cl.^). The maximum cross-section or frontal area of ror. ationally symmetric bodies is S0 = d27T/4. The projected lateral or plan-form area S + (if used as the equivalent of “wing” area) is approximately:

Sf = (0.8)і d in ellipsoids (or airships)

= (2/3)Л d in pointed streamline bodies

where o.8 tt/4. This area is indicative of the non-linear

cross-flow forces, at higher angles. In the case of airships and flying-boat hulls, their volume denoted by V is signifi­cant for performance (buoyancy). Therefore, coefficients are very often vased on the length V ^3 , the area V %; and the moments are referred to the volume V. The ratios of volume in streamline bodies of revolution are roughly as follows:

V = 0.6T d in ellipsoids (airship bodies)

V = 0.5 f d in pointed streamline shapes

In naval architecture, in regard to the displacement of ships, this type of ratio is called the block coefficient, “B.” Thus for bodies of revolution

“B” = V/id2 = .5 to.6 1. LIFT DUE TO CIRCULATION

A streamline body operating at an angle of attack de­velops a lifting or normal force similar to that of a low aspect ratio wing. As explained in Chapter XVIII this force consists of two different components, due to circu­lation and cross-flow.

Theory as presented in (l, a) predicts that in a slender body of revolution, lift develops corresponding to local cross forces per unit length:

dF/dz = q(dSc /dx) sin(2c£) (1)

where dS0/dx = rate of change of cross-section area. When integrating this equation over the length of a complete (“closed”) streamline body, positive lift is found in the forebody and a negative contribution in the afterbody; the resultant lift is always zero. Considering, however, a circular cone with finite length and a blunt base, its lift coefficient based on base or frontal area S, is found to be

C|Lo= sin(2oc) = 2 sinoC cosa: ^ 2oc ^)

This type of lift is due to longitudinal circulation as in wings and/or airfoil sections.

Подпись:Delta Wings. The derivative of the linear (circulation-type) component of lift as in equation (3) is the same as that for small aspect ratio wings, Chapter XVII. In fact, the cone in figure 1, with b = 6 t, can very well be considered to be a delta wing. Experimental results of another family of elliptical cones (with b = 3 t = constant) is plotted in figure 2. At larger values of the length ratio, the same average level is obtained as in figure 1. At the other end of the scale, the aspect ratio increases toward infinity. For example, at^/b = 1, the ratio is A = 2 hz /ЪХ – 2 (b/jf) = 2, which is no longer slender. Modifying equation 6 in Chap­ter III to (12 + (10/A2 ) + (20/A)); and replacing A by 2 ЪЦ, we can write

doC/dCLb = (24 ЪЦ) + (SJl/Ъ) + 1.7 (4)

This function, resulting in dCbb/dcxr = zero at J/h = zero, is verified by test on various slender delta wings. Equation 4 also extrapolates with good results the test results on cones toward_//b = zero.

Подпись:Elliptical Cones. The forces discussed do not correspond to the volume of the body, but to that of the air affected by the body. The cylinder of air deflected by a conical body corresponds to its “span” b (see figures 1 or 2). Therefore, to accommodate cones with cross-section shapes other than circular, it is best to refer their lift to


the area Sb = b. The lift-curve slope of elliptical cones from equations 1 and 2.

dCLb /йофг/2Хтг/180) = ^7360 = 0.0274 (3)

Where C L b is the lift coefficient based on b2 .

Experimental results on a “family” of cones with elliptical cross section shape varying in span/thickness ratio, are plotted in figure 1. Although their lift is between 10 and 20% below the theoretical level, the graph demonstrates the fact that equation 3 applies to “all” types of cones, from slender “upright” elliptical shapes (with t con­siderably larger than b) to slender delta wings (with t zero).

(1) Theoretical analysis of slender bodies:

a) Munk, Airship Theory, NACA T Rpt 184 & 191 (1923/24); also in Vols I & VI of Durand’s “Aerodynamic Theory”.

b) Vandrey, Pitching Moment, Yearbk D’Lufo 1940 p 1-367.

c) Lighthill, Supersonic Body Flow, ARC RM 2003 (1945).

d) Tobak, Derivatives of Cones, NACA TN 3788 (1956).

e) Kaplan, Potential Flow, NACA T Rpt 516 (1935).

f) Sacks, Forces and Derivatives, NACA TN 3283 (1954).

g) Spreiter, Wing-Body Configurations, NACA T Rpt 962.

h) Young, Family of Bodies, ARC RM 2204 (1945).

i) Upson, Airship Analysis, NACA T Rpt 405 (1932).

(2) Characteristics of pyramidal bodies:

a) Flatau, Chem RD Labs, Spec Pub 1-31 (1961); Astia 270250.

b) Paulson, Delta Vehicle, NASA TN D-913 (1961).

c) Ware, Reentry Configuration, NASA TN D-646 (1961).

d) Olstad, Same as (c) at M = 0.6 to 1.2, NASA TN D-655 (1961).

(3) Lifting characteristics of conical bodies:

a) Jorgensen, Elliptical Cones, NACA T Rpt 1376 (1958).

b) Stivers, Family of Cones, NASA TN D-1149 (1961).

c) Flatau, Army Chem RD Labs Spec Pub 1-32 (1961); Astia 270838.

Mach Number. The cones as in figure 1, were tested at a supersonic speed, corresponding to M = 2. The cross-flow velocity on which the lift of a slender cone depends, is w = V sinoc. At the sides of circular cones, this component will be doubled corresponding to the “2 alpha” principle. The perturbation velocity due to the displacement by the cone is approximately (V sin£), where £ = half-vertex angle. For example, for o’ and £ , each equal to 0.1 (corresponding to 5 or 6 degrees) the maximum cross-flow velocity may then be w = 0.3 V. Therefore, the effective Mach number for the experiments as in figure 1, is only a fraction of the number at which the wind tunnel was operating. Ultimately there will be a critical condition, however, where the Mach cone coincides with tha: placed around the longer axis of the elliptical cross section. For M = 2, the corresponding critical cone shape corresponds to ^/b = 3.5 which is obtained above b/t = 6 in the graph. Results on the other family of cones (in figure 2) are shown both for M = 0.6 and = 1.4. The fact that there is a difference (in the order of 10%) in the lift coefficient between subsonic and supersonic speed, may partly be accidental. Within the limitations of theory (really slender cones at small angles of attack) there would not be any difference. The other fact, that none of the values tested comes closer than to 90% of the theoretical level, equation 3, is easily explained on the basis of “round” lateral edges. Evidence presented in Chapter III confirms that 80 or 90% will be a reasonable value to be used in combination with equation 3.

Lateral Forces. The type of cone as in figure 1 was tested for b/t = 6 as well as for t/b = 6. Considering the cone with b/t = 6, to be a delta wing, the point at b/t = 1/6, represents the lateral force deriviative (based on the square of the span b = 6 t)

dCy^ /А/3 = (t/bf dCLb /doc = 0.025/62 = 0.0007 (5) for that particular wing.

(4) Analysis of slender (delta) wings:

a) Jones, Delta-Wing Lift, NACA T Rpt 835 (1946); also J Aeron Sci 1951 p 685.

b) British Efforts, ARC RM 2596, 2819, 3116.

c) Lomax, Slender Wing Theory, NACA T Rpt 1105 (1952).

d) Spreiter, Slender Configurations, МАСА T Rpt 962 (1950).

e) Sacks, Derivatives, NACA TN 3283 (1954), as in (1).

(4) Analysis of non-linear Uft in small A’ratios:

k) Bollay, Zero A’Ratio, ZaMM 1939 p 21; also J Aeron Sci 1936/37 p 294.

l) NACA, Statistical Interpretation, TN 2044 & 3430.

m) J Aeron Sci 1953 p 430; 1954 p 134, 212, 649.

n) Brown, NACA TN 3430; also J Aeron Sci 1954 p 690.

o) The non-linear problem is now claimed to have been solved; see Gersten, Non-Linear Theory, ZFW 195 7 p 276 and Ybk WGL 1958 p 25. No explicit formulation seems to be available, however.

p) For “end-plate” principle, see (72) in the “wing” chapter.


Figure 3. Normal force distribution on a slender streamline body, at oc =10°:

a) as predicted by theory (l, c)

b) as tested (7,b) by pressure distribution.

Load Distribution. As mentioned before, a complete body (with a tail reducing to a point) would have zero lift, in non-viscous fluid flow. The load distribution along the axis of a very slender streamline body is plotted in figure

3. As shown in (l, c) or (7,b) the change in surface pressure due to angle of attack (as against the distribution at oc =0) is theoretically as follows:

a) Along the sides of the body, the minimum pressure obtained corresponds to

Cp = —3 sin2 <x (6)

as in a circular cylinder inclined against the flow (“2 alpha” cross flow).

b) Along the plane of symmetry, the pressure variation due to thickness at oc = zero, is:

A Cp = siri2 oc ± sin(2oc)(dd/dx) (7)

where the first component represents the cross-flow stag­nation pressure. The second component is due to the change in body diameter. The plus sign applies to the bottom, and the minus sign to the upper side. Since the variation of the diameter dd/dx is positive in the fore­body, we obtain a lifting normal force there. In the afterbody, the theoretically expected normal force is negative (down) so that the resultant force is zero. The normal force loading is proportional to (dx/d)(dd/dx), where x = distance from the nose point of the body, and dx = diameter at x.

Подпись:Подпись:— о. З X—1>"

————– C = 00 си————- d = o. 1 5 (

minimum drag coefficient. CD = o.05 smooth

= o.20 rough surface

coefficients CLb = L/qb2 ; Cm£ = M/ql/Д

“Viscous Lift. ” As demonstrated in figure 3, tested pres­sures and normal loads agree fairly well with theory, on the forebody. The corresponding lift is the same as indi­cated in equations 2 and 3 for cones. In the afterbody, theory expects negative loads. In reality, the pressure increase along the upper side of the afterbody, to the centerline pressure as in equation 7 does not take place. At the angle as tested, there is no positive lift in the afterbody. The lift exists in the forebody. However, the negative lift in the afterbody is reduced; the result is a net positive lifting or normal force.

“Edges. ” Although viscosity is the basic reason for the development of positive lift on a streamline body the boundary layer is not necessarily responsible for the de­ficiency of positive pressure at the upper side. In fact, there is “no” accumulation of B’layer material, and “no” separation. Rather, there is a change of flow pattern from that predicted by potential theory, to the real one as in figures 4, 5, & 6. Without friction, there would not be circulation and/or lift. However, the slightest trace of viscosity is sufficient to transform any “sharp” trailing edge into something similar to a stagnation line as in the case of airfoil sections. In the case of a streamline body, we do not have sharp edges. However, skin friction and the absolute refusal of the boundary layer to proceed against a strong positive pressure gradient, makes “edges” out of the lateral sides of the body. All this starts on a streamline body at roughly 50% of the length, where the negative pressure gradient due to thickness along the fore­body and due to longitudinal flow, begins to change into a positive gradient. Therefore, the forebody retains a positive lifting load as approximately described by equa­tion 3.


Figure 5. Pattern of the pair of vortices containing the circulation produced by an ogive-cylinder body (NACA Rpt 1371).

(5) Slender delta wings, experimental:

a) British ARC, RM 2518, 3077.

b) NACA, TN 1468 & 2674, T Rpt 1105.

c) Lange, ZWB UM 1023/5 (NACA TM 1176).

d) Truckenbrodt, 0012 Deltas ZFW 1956 p 236; also Swept and Deltas, ZFW 1954 p 185.

e) Fink, Delta Plates, ZFW 1956 p 247.

The flow pattern around lifting streamline bodies has been investigated by way of pressure distribution (7,b) (8), measuring boundary layer distribution (6), through visual means, such as smoke (7,a, c), and by surveying the wake (7,a) (12).

Important facts are as follows:

a) As shown in figure 6, the flow around the sides begins as in non-viscous fluid. The “2 alpha” type of flow is evident. It should also be noted that there is some flow around the forebody sides, over the top of the body (circulation).

b) The streamlines do not reach the body’s plane, of symmetry at the upper (suction) side, as expected by theory. In fact, there is a line (as in figure 6) beyond which the direction of the flow at and near the surface of the solid, is reversed.

c) While theory may consider the lift of wings and/or bodies, assuming that a cylinder or a sheet of air simply be deflected (downwash), in reality and by necessity, that stream of air starts immediately and vigorously rolling itself up into a pair of trailing vortices of the type as shown in figure 5.

Vortex Pair. As suggested above, it is not necessary or not even justified, to assume that the reversal of the flow as in (b) above, is “separation.” What we find in lifting bodies, is basically the same flow pattern as around slender (low – aspect-ratio) wings. Quoting from (7,b), “as soon as the incidence is increased from zero, a symmetrical pair of spiral vortex sheets form on the lee side.” At small angles of attack, the point along the body where the trailing pair can first be identified, may be at or possible behind the tail end of the body. As the angle of attack is increased, that point moves forward. As shown in (7,d) and (14,c) the location varies as x/d, /<x and it may reach a final position somewhat ahead of the maximum diameter Fig­ure 5 shows the flow pattern in the plane tangent to the base of an ogive-cylinder body. The lateral distance be­tween the pair of trailing vortices is ~ 2/3 of the body’s diameter. The fact that in figure 4, the vortex “span” is only 1/3 of the maximum diameter, indicates that this body (tapering to a short vertical edge) is less effective in producing lift than the cylindrical body above.

(6) Boundary layer on airship models tested: a) Klemperer, LZ-126 (“Los Angeles”); Aerody Inst TH Aachen 193 2.

b) Freeman, “Akron” Model, NACA T Rpt 430 (1930).

c) ARC, British R-101, RM 1169 & 1268 (1929).



Figure 6. Flow pattern observed (7,a) directly at the surface of a streamline body of revolution similar to that of an airship.

Lift = /(Drag). In a closed streamline body with more or less round cross section, there is no geometrcial definition, neither of the effective area, nor of the aspect ratior and/or the effective span. The only method of obtaining information of lift of such bodies, is statistical. Since the type of lift considered, origianally stems from viscosity and boundary layer interference, it can be speculated that it is a function of the body drag. Normal force derivatives (obtained around zero angle of attack, or zero lift) are plotted in figure 7, as a function of the minimum drag coefficient, representing friction (boundary layer thick­ness) as well as the “resistance” of the lateral “edges” of the bodies tested, against the flow around them. Figure 4 presents direct proof for the fact that lift increases with drag produced by surface roughness.

(7) The flow pattern around streamline bodies:

a) Harrington, Origin of Lift of Body, Guggenheim Airship Inst (Akron, Ohio) Pub 2 (1935); also J Aeron Sci 1934/1935 p 69.

b) Spence, Flow Structure, RAE Aero 2406 (1955).

c) Maltby, Vortex Suction, RAE TN Aero 2482 (1956).

d) Jorgensen, Vortex Pattern, NACA T Rpt 1371 (1958).

e) Gowen, Inclined Vortex Wakes, NACA RM A53117.

(8) Pressure distribution on slender bodies:

a) Allen, Influence of Viscosity, NACA T Rpt 1048 (1951).

b) Fuhrmann, Ybk Motorluftschiff Studiengesellshaft, 1911; also ZFM June and July 1910; also ZFM 1911/12.

c) Cole, Bodies of Revolution, NACA RM L52D30.

d) Matthews, Theory and Experiment, NACA T Rpt 1155 (1953).

Appendages, a canopy on top of a fuselage or a windshield cut into the fuselage, also increase the lift-curve slope; see figure 8. A drastic example for this type of interference is shown in figure 9. The lift induced in the fuselage by the disk (simulating a very bluff windshield) is as high as that otherwise produced by an angle of attack between 10 and 20 . The lift is evidently obtained by preventing the increase of pressure along the upper side of the afterbody, thus destroying the negative lift originally produced

Square Cross Section. The lifting characteristics of three bodies differing in cross-section shape, are plotted in fig­ure 10. The left curve slope of these bodies and others are given in figure 11. Physical interpretation of the data is as follows:

a) Plain, smooth and round streamline bodies have little lift, their flow pattern is closest to the ideal, without circulation.

b) Lateral edges particularly in form of square cross sec­tion help to make streamline bodies lifting. Even the edges of polygonal airship hulls are sufficient possibly to double the lift-curve slope.

c) LIFTING CHARACTERISTICS OF STREAMLINE BODIESПодпись: Figure 8. Longitudinal characteristics of a plain fuselage body (11,a) with and without a pilot’s windshield. A full shape of the afterbody, ending in a blunt base, does not directly contribute to circulation. A wide after­body reduces or may even eliminate any negative loading, however, so that the resultant normal force is increased to two – or threefold compared to smooth and round stream­line bodies ending in a point.

Подпись:(Tarabolic,> Bodies. The lift and moment of a series of bodies formed by parabolic arcs are shown in figure 12. Their afterbodies, which reduce in width to lA of the maximum at the blunt base, cause a reduction of the lift compared to that of equation 3. For example, the shapes with span/thickness ratio b/h = 1.0 and 1.5 have dCLb /doc values between 0.005 and 0.008, which is roughly Уа of that of slender cones. As the span/thickness ratio is increased these bodies more and more develop into

(9) Plain and round fuselage bodies:

b) Several Shapes, AVA Gottingen Rpt 1934/03.

c) Engelhardt, Roughness, TH Munich Aerody Lab Rpt


d) Goodman, Fineness Ratio, NACA T Rpt 1224 (1955).

e) Hollingworth, Length Ratio, W’Rpts L-8,12,17 (1945).

f) Muttray, AVA Results, Lufo 1928/29 p 37; as (10,a).

g) NACA, Length Ratio, TN 2358 & 2587: T Rpt 1096.

h) Queijo, Length and. Shape, T Rpt 1049 (1951).

j) Lange, Eight Shapes, ZWB FB 1516 (194); NACA TM


k) Gillespie, M = 1.7 &^/d = 17, NACA RM L54G28a.

CDi = CL71’2/ 54 = °-3

“wings” with lateral edges. It should be noted that cross – section shape, rather than fineness ratio jh (or any other of the parameters used in this analysis) determines lift and longitudinal moment. In fact, at b/h = 3, the lift derivative is already 0.024 which is close to that of equation 3. The moment derivative (about the 0.58 point) is compara­tively constant (around —0.01) and it fits sufficiently well into figure 13.



a) body of revolution

b) with square cross section

c) same body at 45 degrees

The bodies have all the same e^.ze cross – eeotion area, corresponding to s я Q


The coefficients are based on this area. Moments are not reported.


Figure 10. Lifting characteristics of streamline bodies of differing cross section, tested by AVA (10,a).


Figure 9. Flow pattern around, and lift “induced” by a circular disk attached to the streamline body as shown (9,c).


Boundary Layer. In an ogive plus cylinder combination, the afterbody is theoretically not expected to produce any lift, particularly not at small angles of attack. There is consistent evidence (8,a) (14,a, c), however, to the effect that such bodies have linear lift derivatives larger than indicated by equation 3. An explanation may be found in the growth of the boundary layer (displacement) and/or the mass of air within boundary layer and wake added to that originally deflected by the body. The effect seems to be in the order of

Д (dCLb /doc) = (0.0001 to 0.0002) Д/d (8)

In the shapes with cylindrical afterbodies, as in figures 12 and 15, for example, the increment is in the order of 10% of the lift indicated by equation 3. Experimental results are assembled in figure 16.


(11) Airplane fuselages, experimental:

a) Brennecke, Fuselage Shapes, AVA Gottingen Rpt 1935/41.

b) Gruenling, Me-109 in AVA Tunnel, ZWB Dct UM 7857 (1944).

c) Imlay, Yaw Derivatives, NACA TN 636 (1938).

d) Hoerner, Correlation, Ybk D’Lufo 1944, Rpt IA,022.


(10) Fuselage bodies with various cross-section shapes:

a) Square Cross Section, Erg AVA Go Vol II (1923).

b) Bates, Various Shapes, NACA TN 3429 (1949).

c) House, Elliptical Cross Section, NACA T Rpt 705 (1941). e) Letko, Cross Section Shape, TN 3551 & 3857 (1956).



Подпись: в 10 12 14 LENGTH/DIAMETER RATIO









Figure 11. Lift-curve slope of various types of streamline bodies as a function of their fineness ratio.

reference point fcr
moment, at 0.57,1*

diameter d ■ б inch point of reference 0.58 S. = о.0075 і2 = constant


асЇЇЬ/.ісА= O.035

о.90——————- Ф

dC^/dtx, = 0.008



sting ^





= O.027

= о.012 = o.032 – О.013


Figure 12. Lifting characteristics of a series of four bodies (I3,e) with elliptical cross section shape. TN 4362.

(12) Influence of appendages:

a) Jaquet, Fuselage with Ducts, NACA TN 3481 (1955).

b) AVA Gottingen, Windshields, Test Rpts 1934/03 & 1935/41.

Подпись: 0 1 1 ^ о Ю 20 Лм
Подпись: dcyd/dp = 0.021 (о. 02 2)

In the previous section the linear circulation-type com­ponent of the lift of streamline bodies was covered. As the angle of attack and/or the fineness ratio is increased, a second and non-linear component of lift or normal force originates. This component of lift is similar.

Circular Cylinder. Within the range of very small angles of attack, the rear end of a low aspect ratio rectangular wing does not contribute anything to lift. However, as the angle of attack is increased above a few degrees, the component of flow normal to the wing becomes significant in magni­tude. As a consequence, a “second” and non-linear lift or normal-force component develops. The mechanism pro­ducing this component can be better understood, when first considering a long circular cylinder (with aspect ratio A = d/i —^ zero) inclined against the oncoming fluid flow. The experimental results of such cylinders, pre­sented in figure 9, apply to, Reynolds numbers below the critical (which is around R^ = 3(10)5). Disregarding a small component of skin friction drag in longitudinal flow (at small angles of attack) there are only normal forces; and the resultant force is split up geometrically into:

D = N sinot

L = N cosoC (9)

(13) Blunt-basedfuselage bodies:

a) Loving, Boat-Tail Comparison, NACA RM L52J01.

b) See references (9,d)(10,b)(10,e).

c) Goodman, Configurations, NACA TN 2504 (1951).

d) Wolhart, Various Configurations, NACA TN 3649.

e) McDevitt, Family of Streamline Bodies, NACA TN 4234, 4280, 4362.

Figure 16. Experimental results showing the increase (due to viscosity) of the lift-curve slope of cylindrical bodies with their length ratio.

Cross Flow. At the angle of attack oc = 90 , the drag coefficient of the cylinder is

COo =СЛ/9с=СС CO)

based on cross-flow area (.(d). For the cylinders as in figure 17, this coefficient is between 1.1 and 1.2. When the angle of attack is less than 90°, it is now speculated that the normal force corresponds to the cross-flow dy­namic pressure

qw= 0.5 p w2 = 0.5 f (w/V)2 У (11)

where the component of the speed V normal to the cylinder axis is w = V since, and (w/V) = sin2 oc. The velocity component parallel to the cylinder axis, V cosa, does not contribute to the cross-flow pattern around the body. Using now the cross-flow coefficient Cc , the cylin­der’s normal force corresponds to

CN = N/qd^ = Cc sin2oC (12)

Using equation 9 the two components of this coefficient are then

CD = Cc sin3 oC (13)

CL = Cc sineo£ cos<x =Cc sin(2od) (14) Figure 17 confirms these functions fairly well.

усв = 1.1 Sin3 of +.02



Below the critical Reynolds number, see for example in Chapter III of “Fluid-Dynamic Drag.., the cross-flow co­efficient of the circular cylinder is fairly constant (Cc^ 1.2). Above Rd = 4(10)5, cylinder drag coefficients are known to be much lower; at the level of CDo = 0.3. Within the transition range, it must be realized that the “effective” Reynolds number varies with the angle of attack. While at at = 90 , Rj (on diameter) is the proper indication for the flow conditions within the boundary layer, the length of the cylinder would rather be responsi­ble at angles of attack near zero. Experimental results within the critical range, as in figure 18 demonstrate complications as follows:

a) Due to friction and boundary layer growth in tangential direction, the cross-flow coefficient at lesser angles of attack, is in between the sub – and supercritical level; it is in the order of Cc =0.6.



й A WIRE, ARC (a)






oC cos. of






90 ^ 0° Я








b) As the angle of attack is increased, the effective R’num – ber may reduce to below the critical. As a consequence, the cross flow coefficient increases in one set of experi­mental results to Cc = CDo = 1.2.


Figure 17. Lifting characteristics of circular cylinders at Reynolds

numbers below the critical (where C = 1.1) as a function of


their angle of inclination.,




c) Within the range of sufficiently high Reynolds num­bers, the cross flow coefficient reduces to Cc = CDo = 0.2. This very low value is supercritical.



Evidently, in round bodies, the cross-flow mechanism produces the highest lift at higher angles of attack (above oc = 60°) and/or at Reynolds numbers definitely below the critical.

Подпись:Transition. At small angles of attack, there will be a perfectly symmetrical slender-wing-type flow pattern as in figures 4 and 5. It is also well known that at very high angles of attack (say at <x = 90°) and at Reynolds numbers below the critical (say below 3(10)6 on body diameter), the flow pattern is that of the alternating (in side and time) Karman vortex street. Between the two extremes, a transition takes place in two steps. Experi­ments on slender streamline bodies reported in (19) de­scribe the first of these. At a certain angle of attack, the magnitude depending on Reynolds number, the flow pat­tern past the nose of a slender streamline body ceases to be symmetrical. The critical angle may, for example, be 20° (decreasing as the R’number is increased).. Above the critical angle, the flow “separates” from one side of the nose, thus giving rise to a steady yaw moment to the other side. As the flow progresses along the length of the body,

a second vortex forms at the other side; and the “sepa­ration” switches to that side. At the same station, the first vortex detaches itself from the body. The resultant values of lift and drag do not seem to depend on the type of symmetric or unsymmetric vortex formation. The pattern described is still “static” (no fluctuations). It seems, how­ever, that here is the beginning of the second step, the alternating type of vortex street as we know it to prevail at higher angles of attack. Fluctuating values are con­nected with this flow pattern, of drag and lateral forces, and most likely of lift and yaw moments.

Pyramidal Shape. Since cones can be used as lifting de­vices, the delta-type body as in figure 19, having triangular cross sections will also develop lift. The shape was tested both with the flat side forming the top and the bottom. Note that the angle of attack is defined as against the ridge line. The flat top or bottom is evidently more effective, however, than the “V” shaped side since,

a) The lift-curve slope is higher in the flat-bottom con­figuration. The derivative (near zero) is practically equal to the theoretical value, (equation 3, with CL based on projected “wing” area.

b) The non-linear component of lift or normal force corresponds somewhat more to the angle of attack of the flat side than of the ridge line.

Drag. Assuming that the non-linear force really varies as CM2 — sinaoc, an average cross-flow coefficient can be evaluated in the order of Cc =1.0.

Extrapolation to oc = 90° , suggests on the other hand:

CDo = 0.9 for the flat-top configuration

= 1.3 for the flat-bottom shape

The linear component of lift (equation 2) is supposed to reach a maximum at 45 , and to reduce to zero at 90°. Between 40 and 45°, the is some discontinuity; it seems that the flow “separates”, or that the flow pattern changes into the alternating type mentioned above. We may speculate that the circulation type of lift breaks down at this point.

(14) Characteristics of ogive-cylinder bodies:

a) Perkins, Normal Forces at M = 2, NACA TN 3715 & 3716 (1956).

b) Jorgensen, Flow Pattern, NACA Rpt 1371 (1958).

c) Tinling, Pressure and Wake, NASA TN d-1297 (1962).



Figure 19. Aerodynamic characteristics of a basic reentry vehicle body as tested (43,b)(2,c) at 60 ft/sec.

(A) Lift and normal force as a function of the angle of attack.

(15) Characteristics of ducted bodies:

a) Theory, see (62) on “ring wings” in the “wing” chapter.

b) Fletcher, 5 Annular Foils, NACA TN 4117 (1957).

c) Brown, Open-Nose Bodies, NACA T Rpt 808 (1945).

d) Brown, At Supersonic Speeds, NACA W’Rpt L-720 & -728 (1946).

(16) Ganslen, Aerodynamics of the Javelin, Univ of Arkansas Rpt 1960.

Within the range of lesser angles of attack, say to 30 or 40°, the drag is compared to the lift, figure 19. Although there is a considerable amount of drag due to viscous vorticity, the experimental functions show some similarity to that of the theoretical induced drag. Optimum lift/drag ratios of the “vehicles” are found at comparatively small lift coefficients:

L/D = 2.9 at CL = 0.3 for the flat top,

= 2.6 = 0.2 for the flat bottom.

A Javelin is basically a streamline body with extremely high fineness ratio. Typical data as for a type used in competitions, are as follows:

= 100 inch length d = 1 inch f/d = 100 S =1.35 ft wetted area

W =1.75 lb weight

V — 60-80 ft/sec speed x = 38% center of gravity

x = 45% center of area


Lift and normal forces as tested in a wind tunnel are shown in figure 20. Evaluation yields a cross-flow co­efficient between 0.6 and 0.7. Near oc = 0, where the Reynolds number on length, is = 5(10)b, boundary layer conditions must be turbulent, possibly 1 foot from the point. However when operating at a: = 90°, the Reynolds number based on the diameter rather than length is no higher than R^ = 5(10)4. Therefore, the cross-flow coefficient must be expected to increase from the 0.6 level near с* = zero, to between 1.1 and 1.2 at oc = 90°. In fact, at оc = 30°, one of the javelins tested, appeared to have a Cc =0.9. The minimum drag/lift ratio is found (16) at c*r = 25°, in the order of D/L= 0.60. For comparison, equation 9 suggests a value of D/L = since /cosoe = 0.47. The difference must be due to tan­gential forces skin friction and the drag of the grip pro­truding from the otherwise smooth surface. When throw­ing javelins, average angles of attack someshat above 25° seem to be best.

Airship Hull The lateral force characteristics of the bare hull of an airship are plotted in figure 21, where the normal force coefficient is based on the “lateral” area (d-0; subscript Although this hull was only tested up to the angle of sideslip j5 = 20 , extrapolation to 90° is possible in the following manner:

a) Near = 0, the linear (circulation-type) component of the normal force coefficient (on -{b) seems to be

CNo = 0.0015^3° ; dCNb/daC = 0.0015 Ц/d) = 0.012

For the minimum drag coefficient (on b2) CDb = 0.06, the corresponding value in figure 7, reflects the fact that the airship has a polygonal cross section (with a number of longitudinal edges).

b) Equation 2 indicates that the circulation reduces to zero when approaching the angle /3 = 90°. It may be assumed, however, that the circulation-type of lift dis­continues far below this angle as a consequence of separa­tion.

c) The non-linear component is assumed to be as in equation 8. Using the value at j& = 20°, the cross flow coefficient is found to be Cc^Q = 0.35, so that:

А Сд/^5 = 0.35 sin2 oC

At higher angles of attack, we have plotted this com­ponent, assuming that the linear type of lift would be

Подпись: (18) Nose and Tail shape on cylindrical bodies: a) Loving, 4 Different Shapes, RM L52J01. b) Boat-Tailed “Fuselages”, NASA TN D-217 & 950. c) Polharmus, Four Nose Shapes, RM L57F25. Подпись: (17) Characteristics of inclined circular cylinders: a) Relf, Inclined Wires, ARC RM (1917). b) Mustert, Lift and Drag, ZWB Doct FB 1690. c) Bursnall, Critical R’Number Range, NACA TN 2463 (1951). d) See “Fluid-Dynamic Drag”, Chapter III. e) Allen, Cross-Flow Characteristics, NACA RM A50L07.

The value 0.35 (on b>f) represents a cross-flow coefficient on projected lateral area CCo = 0.41, which is very reason­able for the polygonal shape considered, at Reynolds numbers above the critical. However, if one would really have tested the airship hull at/5 = 90°, its proper R’num­ber (on diameter) would have been a little above 105, (instead of 10b, on length when near /в = 0). The cross- flow coefficient would then have been in the order of


seen from below full-scale dimensions:


= 84



= 8.2

* d


= 550



= 0.83 –



= о.05 j

• (4Jt)


= 0.83

* JP

model test (RM 799):

1 to 200 scale

V = 40 ft/sec

Rj * 106

= o.07 bare hull)

Cjj =o.09 minimum

= 0.13 with 7 "cars"


Figure 21. Characteristics in pitch and/or yaw of the hull of the British airship R-38 (ZR-2) as tested (20,b) on a 1/200 scale model.

Drag characteristics of the hull may add to the under­standing of the flow pattern. The coefficient (on d2") is approximately:

CDd = 0.06 within oc= plus/minus 7°

= 0.22 at oc = 20°, almost 4-fold

= 2.90 at oc = 90°, see above

Roughly, the pressure drag and the second component of the normal force, grow in proportion to each other.

The cores of the pair of trailing vortices shown in figures 4 and 5, contain the momentum losses corresponding to the viscous drag of the body., The downwash of the flow field and/or the circulation of the vortices represent the in­duced drag of the body. As explained in Chapter VII of “Fluid-Dynamic Drag”, this type of drag can be expected to correspond to

CD>b _CLb/’TT’

which is the basic equation for wings regardless of their aspect ratio. The slope of the experimental values in part (B) of figure 10, shows fair agreement with this equation. The low drag coefficients near zero lift, are believed to be the consequence of a more laminar flow, particularly along the upper side of the bodies tested.

Equivalent Wing. Considering the square shape in figure 10, its lift-curve slope (around zero angle of attack) is dCLb/do<: = 0.014. In comparison to equation 3 this value corresponds to the ratio

(b,/bf = 0.014/0.0274 «0.5

so that the effective span is found to be bj = v/0.5 b. This span is shown in the drawing of the body. The chord of the equivalent “wing” is estimated on the basis of the tested “second” component of lift, corresponding to

Д CLb = 8 sinz c*

Assuming that a flat plate would produce Д CL =2 sin2 oC, the chord of the equivalent rectangle as shown in the illustration, can then be calculated. In comparison to the body of revlolution (having the same volume) the square shape is seen to have about twice the lift-curve slope, and 3.6 times the non-linear lifting capacity.


Parawings are a form of delta wings in the sense that their planform has the delta slope; however, these lifting surfaces with a parachutelike tension structure are completely different. In this case the wing surface slope is maintained by the balance of forces between the airload on the surfaces and the tension in the supporting structure. These wings have been considered for use for a large variety of tasks from hang type gliders to wings for the recovery of space vehicles (14,a). The flexibility of the structure and its capability of being folded broaden the potential uses of parawings so that a brief discussion of their characteristics is desirable.


Figure 34. Lift-drag characteristics of two-lobed parawings.

Lift Characteristics. The lifting characteristics of para­wings depend on the type of canopy used and the aspect ratio. Typical test results (14,d) are given on figure 33 for wings with both types of canopies. The lift coefficient is based on the area of the flat wing pattern. The most striking difference between the two types of wings is the angle of attack for a given lift coefficient. Much larger angles are needed for the same lift coefficient with the conical wing. In the case of the A = 6 wings, CL* are nearly the same. At A = 3 the CLX of the conical wing is slightly better than the cylindrical wing. At the lower values of angle of attack the conical wing is limited by flutter at the trailing edges. This occurs at lift coefficients below.4 as indicated.

The effective shape of the conical wing results in a large variation of twist between the inboard and outboard sections so that the wing tip is operating at a negative lift at moderate lift coefficients. The twist or washout can be eliminated in the case of the cylindrical wing, which leads to tip stalling as CjLX is approached. The difference in the local angle of attack in the case of the two types of canopies determines the difference in the stalling charac­teristics and the lift drag ratio.

Longitudinal Moment. The variation of the pitching moment is given on figure 33. These data indicate that the wings will have a negative pitching moment at zero angle of attack so that for typical parawing applications the center of gravity must be located well below the wing to achieve trim and stability. In the normal operating range with such an arrangement good positive stability is obtained. However, at both the low and high lift coefficient negative stability is obtained which can lead to an end over end tumbling motion (14,b).


Geometric Characteristics. The parawing can be de­signed in many forms as discussed in (14,b), however, only the two-lobed type illustrated in figure 32 will be considered here. As noted in the figure, the wing has a central keel with two straight or curved booms to form the leading edges. The canopies are generally formed as a conical or cylindrical shape as illustrated, and the booms are spread by a bar that also serves to support the load. The shape of the trailing edge will be determined by the sweep of the flat pattern angle TV 0 and ^ie actual sweep angle of the boom. Although rigid members are considered for the booms and keel, a pension structure can be used to improve the storage characteristics. In this case a balance must be made of the forces to obtain the shape needed flor flight and control.

Lift-Drag Characteristics. At a given aspect ratio the canopy shape is the major factor influencing the lift-drag ratio that can be obtained with parawings, as illustrated on figure 34. For wings with the same aspect ratio the cylindrical shape gives a large improvement in the lift-drag ratio over that obtained with the conical shape. This is due to the improved load distribution resulting from the zero wing twist distribution. The improvement in load distribution almost doubles the lift-drag ratio of the wing, figure 34. With the use of a small degree of washout further improvements in L/Dmax can be obtained along with СцЛ.


As previously noted delta wings are primarily used on aircraft operating at Mach numbers above 1.0. At these conditions their performance is superior to other configur­ations with the result of their selection for many high speed aircraft, including the supersonic transport types. The variable swept wing can be thought of as a delta type when in the fully swept configuration and thus can be included as the one best type for operation at M 1.

Subsonic Compressibility Effects. The performance of delta wings given previously in this chapter is for conditions where compressibility effects are zero, M = 0. The effect of increased Mach number is shown in figure 31 and shows an increase in both the slope of the lift curve and lift drag ratio, which peaks at M =: 1. The variation of the lift curve slope with M is due to compressibility and can be accounted by the Prandtl Glauert transformation such as described in Chapter VII. With this correction the potential lift represented by Kp is increased by compressibility to a greater extent than the vortex lift correction К v. To find Kp Ду the aspect ratio is corrected by the equation

A’ = A /l-M2 (10)

where A’ is the aspect ratio used to read К p and К у on figures 9 and 10, and M is the free stream Mach number. The К factors must also be corrected thus

Kpc =Kp / / 1-м2 (11)

Kvc =KV fM (12)

fM = Vх! + tan2W //3 + tan2A (13)

Comparison of the variation of the lift curve slope calculated with equations (11) and (12) with that measured in figure 31 indicates good agreement.




Подпись: Lift at Supersonic Speeds. To determine the lift of delta wings at M > 1 the leading-edge suction analogy is also used. Here the constants of К p and Ky can be determined based on a linerarized theory (12,d) with the result that
Подпись: Kp = ТГА/2Е (14)
Подпись: and Ky = Tf[(16-(A/3 )Z)(A2 + 16)/Z/16E2 (15) where E = elliptic integral of the second kind where modules = [1 ~( cotAT p )2]/2. In this case /3 = /мї - 1. In (12,d) comparisons of the calculated results of equations (14) and (15) with test data indicate good agreement.

Figure 33. Lifting and moment characteristics of two-lobed parawings, A = 3 and 6.


The application of flaps to obtain high lift and control for delta wings requires consideration of the differences in flow produced as a result of the leading-edge vortex. As discussed earlier in this chapter, the leading-edge vortex increases the lift which is non-linear with respect to the angle of attack. Further, the flow produced as a result of the vortex changes the pressure distribution on the wing, so that it might be expected that the effect on lift of a flap deflected into the vortex would be different than if deflected away from the vortex. Since the vortex is influenced by the shape of the leading edge, the effectiveness of flaps will be different for wings with different types of cross sections.

There are many different types of flaps that can be installed on delta wings, such as plain, split and slotted types. Because of the limited span it is generally necessary to use these flaps to obtain high lift and control ;.n pitch and roll. Such flaps are called elevons.

M :r 87.7





Figure 27. Delta wing body combination with plain flaps.

Plain Flaps. The characteristics of plain rectangular flaps attached to the trailing edge of a round leading-edge delta wing are shown on figure 3, Chapter IX. On the basis of flap area to total wing area, the effectiveness of the flaps would be predicted from figure 2 of Chapter IX which was developed from test of flaps on straight wings. The use of flap area to wing area to predict the effectiveness is further confirmed by tests of a large model delta wing in the 40 x 80 foot wind tunnel (13,a). A delta wing with sharp leading edges and a plain type of flap and a fuselage was tested, figure 27.

Flap Effectiveness = (dCL /dot) / (dCL /d6 ) (6)

is close to that predicted for this type of flap on figure 2 of Chapter IX. Although the wing tested has sharp leading edges, a comparison of the lift with the theoretical prediction indicated the slope of the lift curve is low. The low lift curve slope does not appear to change the flap effectiveness and is probably caused by the central body.

Split Flaps. Tests of 60c split flaps on a delta wing with an aspect ratio of 3.03 are given on figure 28. These tests (13 ,b) were done in conjunction with spoiler flaps of the same size to be used together as dive breaks or separately for increasing lift. The flaps were tested at various chordwise locations on the wing for different length spans and flap chords. The split flaps were also tested in the vented and non-vented configurations as indicated on figure 28.

Based on equation 1 of Chapter V the flap effectiveness can be measured in terms of (C^ /C)n for flaps on rectangular wings. For such wings with the same type of section test in conjunction with the above data, the data of (13,b) indicates n would be 1.03. As above, the flap effectiveness ratio is found in terms of the area ratio. This indicates for delta wings with split flaps the effectiveness is

dS/dct’ = (Sf/S)’93 (7)

The test data of (13,b) also indicated that if the flap is vented the n would increase. Further, the inboard flaps are more effective than those outboard when located toward the trailing edge of the wing.

The effect of deflecting split flaps located on the upper surface of the wing is given on figure 27. The effectiveness for decreasing lift is the same order of magnitude as for increasing lift with the flaps located on the lower surfaces of the wing, speed breaks are obtained without changing the basic lift slope of the wing.

Double Slotted Flaps. Because of the high angle of attack needed for landing with delta wing aircraft and the low span, double slotted flaps have been investigated (13,d, e). Typical results are presented on figure 29 for a wing body configuration with double slotted flaps extending to the.666 spanwise station. The flaps were investigated with a large and a small intermediate foil. Up to flap angles of 45 ° both configurations had nearly the same perform­ance at zero angle of attack. This performance improves



with flap angle for the configuration with the larger foil for Да: =* 6° . Above this angle the payoff in

ACl_ is reduced with flap angle until at Cux the 6$ = 45u is best. It is observed that the wing with Sf = 45° has the same lift curve as the basic

configuration up to the stall angle.

For the wing with the double slotted flap with an adequate intermediate foil the flap effectiveness can be predicted based on the flap type, area ratio and the deflection angle, using data such as given in Chapter V. It therefore appears that the performance of flaps on delta wings can be predicted with good accuracy based on the characteristics of the flaps determined on straight wings.

Part Span Flaps. Tests (13,c) were of part span flaps on one side of a delta wing, figure 30, to find the effectiveness as a function of spanwise location. From the data given it can be concluded that the effect of the location of the flap spanwise is of secondary importance and the lifting effectiveness is proportional to the span fraction. Regardless of flap-span fraction, use the value of b/c as the aspect ratio responsible for the flap lift curve slope. For the configuration in figure 30 that ratio is Af = b/cf = 9, and the “lift angle” is

dot /dC(_ =11 + 20/9 = 13.2° (8)

so that dCL I dot = 0.076. The fraction of the ‘‘half”

span flap is actually 0.9/2 = 0.45. The lift due to flap deflection may thus be

dC^b /d6 = 0.076(Sf/b2) = 0.076(0.45)/A (9)

The result of 0.0038 is somewhat larger than the tested value (0.0081/2.31 = 0.0035).

Aileron. The half and/or part-span trailing-edge flaps of the delta wing in figure 31 can conveniently be used as ailerons. Of course, in regard to their moment about the longitudinal axis, there is a difference between outboard and inboard flaps. The roll-moment contributions listed in the illustration yeild arms “y” as follows: outboard and inboard flaps. The roll-moment contributions listed in the illustration yeild arms “y” as follows:

y/b = 0.0004/0.0045 = 0.1; (0.12)

= 0.0012/0.0040 = 0.3; (0.33)

= 0.0016/0.0081 = 0.2; (0.25)

For comparison, the purely geometrical ratios are added in parentheses. Tentatively, therefore, the moment arms may be considered to be between 0.8 and 0.9 of the geometrical ones.

In examining the rolling moments as determined from test, the difference between positive and negative aileron deflection is small at the lower angles of attack. Further, no significant changes are noted between wings with round or sharp leading edges. This indicates that the leading-edge vortex does not influence the results in this range.


Figure 29. Double slotted flaps on delta wing airplane.

Antisymmetric Deflection. In a conventional, more or less straight wing (with higher aspect ratio) the anti­symmetric deflection of the ailerons produces lift dif­ferentials in each wing panel corresponding to a lift curve slope based upon Vi the wing’s aspect ratio. Based on the results of (13,c) the use of up and down ailerons will double the rolling moment shown on figure 30, deter­mined for one side of the wing. At the lower angles of attack the yawing moment is negative with up or down flap deflection switching at angles above 10 degrees. This change will effect the flying qualities of the aircraft thus requiring special design consideration.




.0045 о 0041

о 0081

.0020 о 00’20





Figure 30. Part span flaps on a large delta wing.

(14) Parawings:

a) Rogallo, Flexible Wings for Transportation.

b) Sleeman, Parawing Aerodynamics, A&A Engineering, June 1963.

c) Mendenhall, Aero Characteristics of Two-lobed Para­wings, NASA CR-1166 and J of А/С, Nov, Dec 1968.

d) HIGH LIFT AND CONTROL DEVICESHIGH LIFT AND CONTROL DEVICESПодпись:Polhamus, Experimental and Theoretical Studies of Parawings A = 3 and 6, NASA TND-972.


The Leading-edge vortex formed by a delta wing proceeds downstream as a tightly rolled tube illustrated in figure 3, until it becomes unstable and experiences an abrupt expansion, vortex bursting. At the lower angles of attack the abrupt expansions occur downstream from the trailing edge. As the angle of attack and/or aspect ratio of the wing is increased the vortex bursting moves upstream and intersects the wing trailing edge. With further increases of A and/or o’ the breakdown moves forward, ending at the vortex of the wing. Although there are two types of vortex breakdown (12,a) no attempt is made here to find differences on the wing lift.

Effect on Lift. When the vortex breakdown occurs aft of the wing trailing edge, the vortex lift predicted by equations 1 and 3 continues to increase with angle of attack. This occurs because the vortex reattaches as required by the leading-edge suction analogy. As the vortex bursting position approaches the wing trailing edge reattachment fads and the lift is reduced. This lift reduction occurs when the vortex is slightly aft of the trailing edge of delta type wings as the vortex producing the lift is helical in nature, requiring area for full recovery of lift (12,f). The reduction of lift occurs as shown on figure 25 as the vortex breakdown moves forward on the wing preventing reattachment until the bursting at the vortex of the wing. At this point the vortex lift is zero At this point it should be noted that tests (12 ,g) show blowing near the wing leading edge tends to prevent vortex breakdown. This allows the lift to follow that predicted by the suction analogy (5,a).


Figure 25. Leading edge vortex breakdown position for sharp edge delta wings.






Figure 26. Limits of angle of attack for vortex breakdown or intersection.

Vortex Breakdown Position. The angle of attack and

wing aspect ratio influence the location of the vortex breakdown position as shown on figure 26. The position for the vortex bursting has been obtained from flow visualization techniques, such as smoke (12,a) and is a good indication where the reduction in vortex lift will occur, and the extent.

Slender Wings. In the case of very slender wings, low aspect ratio, the vortices on either side will make contact or have an asymmetry before vortex breakdown occurs. In this case the height of the vortices above the wing are different producing a difference in lift that causes a strong rolling moment. This results in another limitation in lift. The conditions for the contact have been determined from rolling moment data (12 ,d) and are given on figure 26.

(13) Flaps on delta wings:

a) Corsiglia, Large-scale Test Low A Delta Wing, TN D-3621.

b) Holford, Split Flaps on 48° Delta Wings, A = 3.03, ARC R&M 2996.

c) Hawes, Constant-chord Ailavator on a Triangular Wing, NACA RM L51A26.

d) Brown, A = 1.85 Delta Wing with Double Slotted Flaps, NACA RM L56D03.

e) Croom, Thin 60° Delta Wing with Double Slotted Flap, NACA RM L54L03a.

f) Spencer, Longitudinal Control Delta Wings with Nose Deflection and Trailing-edge Flaps, NASA TN D-1482.

g) Thomas, Wing-tip Controls on Delta Wings, ARC R&M No. 3086.

h) Koenig, Delta-wing with Mid Chord Flaps, NACA TN D-2552.


Подпись: A (I f CQ99//C0J

Changes in the lifting characteristics of delta type wings due to changes of geometry from the simple shapes considered in the previous section can only be determined by test. Considerable test data is available so that these effects including changes in planform shape, leading-edge radius, camber and control devices can be found and compared with the lift characteristics of the simple wing found by the modified theory.

Aspect Ratio. The effect of changes in aspect ratio sharp-edge delta wings based on low speed wing tunnel tests is given on figure 13. Since the potential flow lift term can only be estimated by determining the slope at CL= 0 and in accordance with equation 1 it varies as sin of cos2 oc, comparison with the calculated values are made based on the total lift. The agreement between the calculated total lift based on equations 1 and 3 and test data is good for the entire range to A = 2.3, and for angles of attack below where vortex bursting might be expected. The increase of the slope of the lift curve with aspect ratio and angle of attack predicted by the modified theory (5,a) and test data is thus confirmed.


Figure 14. Planform shapes of slender delta wing tested, including ogee, gothic and delta types.


.025- .020-

Л" .015-

OB 09 10 ~U І2 В І4

Figure 15. Potential flow lift of ogee, gothic and delta wings collapsed to a single line.

Planform Shape. The effects of changes of planform shape from the simple delta wing on the potential flow and vortex lift coefficients has been found for a series of sharp leading-edge delta type wings for A from.75 to 1.25 (6). The shapes evaluated include ogee, gothic and delta wings, figure 14. The test data for these wings were collapsed into a single line (7) for both the potential flow and vortex lift components.

Potential Flow Lift. A single line for the potential flow lift curve slope of the series of wings was obtained, figure 15, plotting dC /do as a function of the factor A(1 + C99 /C0). By modifying the aspect ratio by the above factor the planform shape effects are normalized more in line with theory. Since the potential flow lift curve slope can only be estimated from test data at zero CL the data given on figure 15 applies strictly for low angles of attack. To find the slope due to changes in angle of attack the data should be modified by the factor sin or cos2a. The corresponding theoretical line for the potential flow lift calculated from equation 1 and figure 9 is also shown on figure 15. This curve indicates a loss of lift in comparison to the theoretical which increases with aspect ratio.

(7) Delta wings experimental characteristics:

a) Fink, Zeit Flug Wissenschaft 1956 p 247.

b) British ARC, RM 2518 & 3077.

c) NACA, Tech Note 1650.

d) Hall, Wing-Body Configurations, NACA RM A53A30.

e) See also (6).

f) McKay, Reversing Triangular Wing Body, NACA 51H23.

g) Lange, 0012 Wings, ‘ZWB Rpt UM 1023/5; NACA TM 1176.

Подпись: (5)Vortex Lift. The vortex lift element of delta type

slender wings can be correlated into a single line, as shown on figure 16, by modifying the normal force slope using a factor which depends on the span at a semispan from the trailing-edge. This factor

f(S) = bVb

Подпись: Figure 17. Lift curve slope of a series of delta wings showing effect of leading edge condition and comparison with calculated values. where b’ = the span at a distance b/2 from the trailing edge. For the range of aspect ratios considered good cor­relation is obtained when the corrected slope is plotted as a function of o(c /А, figure 16. At /А above 20 the correlated data is below that calculated by theory equation 3, figure 10, indicating the vortices have started to break down in lift effectiveness.








Circulation Lift – Leading-Edge Shape. The potential flow or lift produced by circulation can be estimated from test data of the complete wing by finding the slope of the lift curve at zero angle of attack. Although this can only be done at low angles of attack, it is possible to find the effects of leading-edge radius at least in this range. Shown in figure 17 is the lift curve slope as a function of aspect ratio for different types of leading-edge radius. Also given is the theoretical varation based on the leading-edge suction analogy. In the aspect ratio range up to 1.5 the test data splits the estimated value. While at the higher values of A the experimental slope is above that calculated. Wings with sharp leading edges appear to have higher lift coefficients than wings with round leading edges. It appears, therefore, that part of the lift loss due to tip shape may be associated with circulation lift. We tend to discount this, however, due to the general inaccuracies in the data and slope estimation.













10 15 20 25 50 55


Figure 16. Vortex lift element of delta type wings correlated into a single line.



Leading-Edge Shape. The leading-edge shape, which can also be considered to be the tip shape of delta wings, has a large effect on the lift produced by the wings. It appears that the proper formation of the vortex in conjunction with the spanwise flow is essential to the production of lift. As previously noted, the formation of the vortex and estimation of its lift is based on the assumption of sharp edges as illustrated in figure 8. Any deviation in shape from the assumed sharp edge can be expected to produce a reduction in the lift. From the flow patterns observed (8) it appears that the effect of leading-edge shape mainly effects the vortex lift. This is difficult to confirm as the vortex and circulation lift components are not measured separately.

Vortex Lift – Leading-Edge Shape. Because of the non-linear shape of the lift curve of delta wings, C L is examined at two different angles of attack, 10 and 20 degrees, as a function of aspect ratio, figure 18. The test data is divided up into that for sharp or round leading wings and is compared with the lift calculated by equations 1 and 3. The available data on sharp leading – edge wings shows good agreement with the calculated values at both angles. The wings with round leading edges have a marked reduction in lift with increasing aspect ratio. This is especially true at the higher angles and aspect ratios above 2.0 where the lift drops off with increasing A. For instance at A above four the lift curve slope begins to decrease when taken at oi = 10 , figure 18. This is

due to vortex breakdown as discussed in the next section.

dihedral angle plus an angle at the leading edge of 17 degrees appears to be the most effective way of improving lift.



Подпись: Figure 19. Variation of lift at a = 10° and 20° as a function of camber.

Camber. To obtain the desired characteristics of delta wings at the high speed flight condition camber may be used, (9,a, b). In this case camber was used to obtain design lift coefficients as high as.075 with the achieved value being 85 to over 100% of the design value. The lift characteristic of the wing with camber appears to follow the uncamber wing with an increment equal to the design value. For wings with an ogee planform the use of camber increased the lift drag ratio (9,b) and has a large influence on the position of center of pressure, depending on the chordwise distribution.

In (9,c) a series of wings of varying camber with an aspect ratio equal to 1.0 were tested to determine the non-linear effects, figure 19. The CL at of = 0, 10° and

20° show that the local angle at the leading edge has a large effect, especially over 40 degrees. The inverse

(8) Characteristics of sharp-edged arrowhead wings:

a) Buell, Flat-top Configuration, NASA Memo 3-S-59A.

b) Treon, Hypersonic Configuration, NASA TM X-364

c) Bartlett, Edge Shape, J Aeron Sci 19-55 p. 517.

d) Shanks, Six Plates, NASA TN D-1822 (1963).

e) Jorgensen, Cones with Plates, NACA Rpt 1378



Double Delta Wings. The double delta wing is used for high speed airplanes for reducing the large rearward aerodynamic-center shift that occurs between subsonic to supersonic flight. Examples of double delta wing con­figurations are shown on figure 20 for supersonic transport type airplanes (10,d). This shift is obtained because of the change in aerodynamic loading from the rear delta at high speeds. Even with the use of the double delta con­figuration it may be necessary to shift fuel longitudinally to maintain the desired stability.

The effect of the use of a double delta wing planform on lift characteristics can be determined based on the leading-edge suction analogy (5,c). For the case where a glove or second delta is added to the basic wing the factors in the lift equation Kp and Kv are given as a function of the sweep angle of the leading edge, figure 21. For this case the vortex lift will increase with an increase of the first wing delta angle while the potential flow lift decreases. This is probably due to the increase in length of the leading-edge vortex and the effective decrease in aspect ratio due to the leading delta. The net result of this is a decrease of total lift compared to the basic wing, figure 22.


Подпись: (9) Camber effects of delta type wings: a) Dobson, Cambered Wing of Slender Ogee Planform, ARC C.P. 778. b) Earnshaw, Low Speed Tests Series of Cambered Ogee Wings, ARC C.P. 775. c) Squire, Non-linear Lift of Slender Wings, ARC C.P. 924. d) Smith, Conical Camber to Produce Flow Attachment- Theory, R&M 3289. e) Dollyhigh, Aft Tail Fighter with Cambered, Uncambered Wings & Uncambered Fuselage, NASA TM X-3078.

The experimental data (10,e) given on figure 22 shows that nearly the same lift is obtained with the double delta wings as the basic wing. Although the level is below the theoretical values, the improvement is probably due to the delay of the vortex lift breakdown to higher angles. Thus, the addition of the glove does not appear to reduce the wing lift. The choice of the combination of double delta wings will thus depend on the center of pressure shift needed in the design and the lift drag ratio, see figure 31.



Figure 21. Potential flow and vortex lift factors, Kp and К v for 65 ° basic delta wing with glove.

Wing + Bodies. As shown in Chapter XIX on the lift characteristics of streamline bodies, vortices are formed that are similar in nature to those produced with delta wings. These vortices lead to the 2 ex’ flow on the body and in the vicinity of the wing root with the result that a strong interaction is encountered. Thus, in determining the characteristics of the wing and fuselage it is necessary to consider:

a) effect of the wing vortices on the wing

b) interference of the wing vortices on the fuselage

c) effect of the fuselage vortices on the fuselage

d) interference of the fuselage vortices on the wing.

The influence of the fuselage on the lift curve slope of delta wings is given on figure 23 for A = 1 and 2. The 2 of cross flow of the body reduces the lift on the wing body combination. This is further confirmed by the two dimensional data of (11 ,b) which shows the effect of wing position. As the wing moves forward the lift increases as shown approaching the theoretical value for the wing alone. This would be expected as the body vortices influence a smaller portion of the wing as it moves forward. In (11) a theoretical procedure for calculating the lift of the combination is given which includes consideration of the fuselage vortices and gives good agreement with the experimental data. In this theory the wing is replaced by a continuous vortex distribution. The trailing vortices are then located in sheets inclined at an angle of Ofoo with respect to the wing plane. Non-linear lifting surface theory is then used to solve for the effects of these vortices on the wing (1 l, a).


Figure 22. Lift coefficient for wings with and without double delta planforms.


(10) Double Delta wings:

a) Freeman, Low Speed Flight and Force Investigations NASA TN-4179.

b) Hopkins, Cranked Leading-edge Wing-Body Combina­tions M = .4 to 2.94, NASA TN D-4211.

c) Подпись: (11) Wing-body combinations: a) Otto, Non-linear Lift and Pitching Moment Coefficients, J of А/С, Aug 1974. b) Hall, Lift, Drag and Pitching Moment of Low-aspect- ratio Wing and Bodies, NACA RMA53A30. Corsiglia, Aircraft Model Double-Delta Wing Longi­tudinal and Lateral Characteristics, NASA TN-5102.

d) Boeing Document 1971.

e) Wentz, Sharp-edged Slender Wings, AIAA Paper 69-778.

Подпись: BODY WITH A = 2 WINGCONFIGURATION VARIABLESCONFIGURATION VARIABLESCONFIGURATION VARIABLESleast as important in the aspect ratios tested for producing lift as the leading edge. It should be noted, however, that the simple conclusion as in (8,d), whereby a wing with zero trailing-edge span may be expected not to produce any circulation, does not appear to be correct. Regarding the second non-linear component of lift, the point-last delta shape is again seen to be the least effective. This result seems to be a confirmation of the end-plate principle presented above. In the shape considered, the lateral vortex sheets are canted in a direction reducing their effectiveness.



Figure 23. Wing body combinations for delta wings of A = 1 and 2.


Delta – Point Flying Last. Delta wings are usually thought of and designed as flying with their vortex point first. Results of such a shape tested in the point-last direction are presented in figure 24. It is very interesting to see that the point-last plate has the smallest linear lift-curve slope, although it presents a “perfect:” leading edge to the stream. In other words, the trailing edge is at

(12) Vortex breakdown on delta wings:

a) Wentz, Vortex Breakdown on Sharp-edge Wings, AIAA Paper No. 69-778 and NASA CR-714.

b) Earnshaw, Vortex-breakdown Position-Sharp-edge, ARC, C. P. No. 828.

c) Lambourne, Breakdown of Certain Types of Vortex, NPC Report 1165.

d) Polhamus, Vortex-lift by L. E. Suction Analogy, J of Aircraft, April 1971.

e) Snyder, Theory of the Delta Wing, Wichita State U., AR66-4.

f) Lawford, Sharp-edge Delta Wing-Position L. E. Vortex Breakdown, RM 3338.


The flow characteristics around a delta wing are influ­enced by. the leading edge angle, cross section shape and angle of attack. These characteristics determine the position and strength of a pair of vortices that start at the apex of the wing. The vortices formed are descri bed as leading-edge vortices and are located on the top surface of the wings, the core proceeding downstream 😮 the trailing-edge at an angle slightly greater than the heading – edge angle of sweep. Aft of the trailing-edge the vortices continue downstream in the direction of flow at a downwash angle proportional to the lift. These vortices exert a large influence on the lift characteristics of the delta wing, especially at the higher angles of attack.

With the rolling-up process there is a strong lateral flow toward the edges as indicated in figure 3. A secondary separation and the formation of another small vortex is developed from this flow. The direction of rotation of this vortex is opposite to the main leading-edge vortex, figure

2. In the main leading-edge vortex there is a progressive reduction of the total pressure and increasing intensity of the edge vortex sheets with increasing angle of attack.

The pressure distribution across the span is shown in part (B) of figure 4. Most of the normal force is produced in the outer 1/2 or 2/3 of the semispan on the upper surface. This type of pressure distribution is influenced by the tip vortices as well as the normal circulation flow on the wing.

FLOW CHARACTERISTICSFLOW CHARACTERISTICSLeading-Edge Vortices. The leading-edge vortices of delta wings have been observed and measured by several investigators (1). Many techniques including tuff surveys, oh film studies, condensation trail photographs and water tunnel studies have been used. Based on these observa­tions the vortex flow pattern for a typical wing is illustrated in figure 3. Vortex sheets are shed off the leading-edges and blow back over the upper surface rolling up to form a pair of stable vortices. These vortices appear on top of the wing and increase in intensity downstream from the apex. By the time the vortex sheet reaches the wing trailing-edge it is fully developed. The pressure differential between the upper and lower surfaces causes the flow velocity and the formation of the leading-edge vortices. As would be expected the pressure difference increases with angle of attack with a resultant increase in the strength of the vortices. As the apex angle of the delta wing decreases, a reduction of aspect ratio, the tip vortices on either side will interfere. The tip vortices on the upper surface of the wing proceed downstream at a downwash angle corresponding to the lift of the wing until they break up.

Figure 2. Characteristics of arrow and diamond shape delta wing modifications.

Flow Pattern. Details of the flow pattern around the edge of a slender delta wing operating at a 15 angle of attack were determined from (l, a) and are given in figure

3. The pressure survey at the 42% root chord show a strong flow around the edge of the wing due to the pressure difference between the upper and lower surface. The vortex sheet originating from the edge rolls up in a large single vortex core. This is basically the same type flow encountered with any other wing. The difference is due to the fact that the lateral edges are long in comparison to the span of the slender shape wings considered.

(1) The flow pattern of slender delta wings:

a) Fink, Experiments on Vortex Separation, R&M 3489.

b) Lee, Delta Wing Flow, ARC RM 3077 (1958).

c) Bird, Flow Visualization, NACA TN 2674 (1952) & TN D-5045.

d) Earnshaw, Structure of a Leading-edge Vortex, R&M 3281.

e) Peckham, Series of Uncambered Pointed Wings, R&M 3186.

f) Poisson-Quinton, Water Tunnel Visualization Vortex Flow, A. A. June 1967.



Figure 3. Flow distribution of a sharp-edge delta wing showing
primary and secondary leading-edge vortices.

Vortex Pair. The pair of leading-edge vortices are located with respect to the wing as shown on figure 5. The height of the core above the wing increases with angle of attack, but the spanwise position decreases. Aft of the wing the core of the vortices achieves a downwash angle equal to approximately 0.5 or. Theory does not predict the location of these vortices with any accuracy (2).

angle of attack increases further, the bursting point moves toward the apex. Although the vortex core breakdown represents an abrupt increase of the core diameter with increased dissipation, it does not lead to a complete absence of vortex type flow. However, when the vortex bursting takes place at the wing trailing-edge, it can be expected that the vortex lift increment will decrease with further increases of the angle of incidence.


Vortex Breakdown. The leading-edge vortex remains stable along the wing surface and proceeds downstream until bursting takes place (3). At bursting the vortex becomes irregular and influences the flow pattern. With a further increase of angle of attack, the vortex bursting takes place on the wing starting at the trailing edge. As the [133]


Ы – 15° – 0.6


Figure 4. Pressure survey of flow pattern in the vicinity of the tip of a delta wing, including spanwise load survey.

Lift Variation. The typical experiment results (4,h) of the variation of the lift and moment of a slender, sharp edge, flat delta wing is given on figure 6. As a result of the potential flow and vortex lift created by the vortices described above, the lift curve is non-line a]’ as is illustrated. The slope of the lift curve increases with angle of attack until vortex bursting takes place, after which a stall takes place with a corresponding reduction of lift. The angle and chordwise location for the vortex bursting is also given on figure 6. The vortex bursting was determined from flow observations. The slope of the moment curve is negative and linear up to the stall angle.

Подпись: BASED ON DATA (4,c) Ch = °-5 b/Cx = 0.5 A =1.0 Подпись:The initial slope of the lift curve at zero angle of incidence represents the linear portion of the lift curve and is equal to the potential or circulation lift of the wing. This portion of the lift curve is the same as was observed in the case of low aspect ratio wings and can only be estimated at zero angle of attack from the experimental data. Above this angle the potential flow lift follows a sin cosa oC function indicated by theory as discussed in the next section.

The experimental lift curve shown in figure 6 also contains the component or lift which is produced by the leading-edge vortices. This lift component is referred to as vortex lift and follows an angle function corresponding to sin2 oL cos or until there is an apparent limit due to vortex breakdown. The vortex lift increment is illustrated on figure 6 and can be calculated for sharp-edged flat wings as shown in the next section.

60 ° Delta Wings. The characteristics of a larger aspect

ratio delta wing are given on figure 7. This wing is a flat plate with rounded leading-edges with zero camber and A = 2.31. The results shown in terms of the normal force coefficient indicate a maximum lift at ot = 33° dropping to a value of.88 at or = 45° .

Lift to oC = 90°. The lift coefficients as in figure 7 sooner or later reach maximum values beyond which the flow separates from the suction side. However, as shown, after an appreciable drop there is still lift at angles of attack between 20 and 40° corresponding to lift coefficients between 0.7 and 0.8. Lift at still larger angles, to 90° , is the result of pressure at the lower side of negative pressure within the separated space at the upper side and as a consequence of some circulation. This type of flow pattern in larger A’ratios at larger angles of attack, characterized by an alternating vortex street, is different from the cross-flow type described above. Basically, we may consider lift at angles near 90° to be a component of pressure drag. To estimate the variation of lift and normal force to or = 90° we assumed, therefore, that = CD = 1.2 (for a flat plate) and reduces as the function 1.2 sin2or. The normal force variation for the entire wing is given for the range based on this variation. At intermediate angles (between 30 and 60°) there is evidently some additional component of lift, other than that due to drag, due to circulation and to deflection of the flow field including the viscous wake.

The history of lift (or normal force) is reflected in the center of pressure point, shown in the upper part of the graph. The center of pressure is first located at x/J? = 0.56 or 0.57. On account of the non-linear component, the point moves forward as far as to 0.54 of the chord. At ot =90 , the center of pressure will necessarily be at

the geometric center of the delta plate, at 2/3 of the length. As the “wing"’ stalls (between 30 and 40^ ) the pressure point moves aft, roughly 10% of the length of the plate.




Figure 6. Typical variation of lift and moment coefficients of a sharp-edge delta wing, including vortex bursting location.

The theory for calculating the lift characteristics of delta wings has received considerable attention (2) without the development of an exact solution. The difficulty appears to be the determination of the location of the leading edge vortices. Further, problems due to separation such as might be developed about a round edge, further compli­cates finding an exact solution. A procedure has been developed, however, for calculating the potential flow and vortex lift components for sharp edge flat plate delta wings. This procedure (5,a, b,c, d) is based on a leading edge suction analogy and is very effective for finding the lift components. Although it is limited to sharp edge type wings this procedure will be used as a basis for determining the lift of delta type wings. The effects of changes due to planform shape, edge radius, flaps, control devices and bodies not considered by the leading-edge suction analogy are then determined based on experi­mental data.


Figure 7. Characteristics of a larger aspect ratio delta wing to high angles of attack.

Подпись:Since К p is dependent on the planform it has been found by a suitable lifting-surface theory (5,3) as a function of aspect ratio for delta, arow and diamond shape wings and is given on figure 9. The lift coeeficient due to potential flow is then calculated using equation 1 and a value of К p from figure 9, which depends on the planform of the wing.







Figure 8. Leading-edge flow conditions comparing theoretical and actual conditions.



Potential Flow Lift. The potential flow lift component

of sharp edge wings is determined assuming no leading edge suction can be developed, since a Kutta type condition is assumed to exist (5,a), figure 8. It is further assumed that the flow reattaches downstream of the leading edge vortex. Although the potential flow lift is reduced by the assumed loss of leading-edge suction, which should be changed to the vortex lift term, it is retained for convenience. Without leading-edge suction the resultant force for plain wings is the potential flow wing normal force which depends on the strength of the circulation. The strength of the circulation about the lifting surface requires the velocity normal to the wing induced by the complete vortex system to be equal to V sin of The velocity of the flow is thus parallel to the wings satisfying the Kutta condition at the trailing edge. By this method a coefficient Kp is determined which is related to the potential flow lift coefficient CLP by

CL ~ К p sinof COS Q’ (1)

Equation 1 becomes for small angles

CL = Kpof; dCL/dtf = .0174Kp (2)

Figure 9. Potential flow lift constant as a function of planform parameters

(5) Theoretical Analysis:

a) Polhamus, Vortex Lift Based on Leading-edge Suction Analogy, NASA TN D-3767.

b) Polhamus, Application of Leading-edge Suction Anal­ogy for Delta Wings, NASA TN D4739.

c) Polhamus, Charts for Predicting Characteristics of Delta Wings, NASA TN D-4739.

d) Polhamus, Predictions of Vortex-Lift by Leading-edge Suction Analogy, J of Aircraft, April 1971.

(6) Experimental investigation of “parabolic” planforms:

a) Peckham, Series of Pointed Wings, ARC R&M 3186.

b) Spencer, Series of Planforms, NASA TN D-1374 (1962).

c) Kirby, “Thin Slender Wings”, Letter report to Hoerner.

Vortex Lift. The flow developed by the leading-edge vortices on the top surface of the wings results in the development of the vortex lift component of lift. This lift component is generated by the downward momentum produced by the flow over the wing leading-edge from the pair of counter-rotational vortices above the wing, as is illustrated on figure 3. The suction peaks associated with the leading edge vortices give an increase in the lift curve slope with incidence and so causes the characteristic non-linear lift curve of slender delta type wings.

Theory. There have been many attempts to develop a theory for calculating the position of the leading-edge vortices on a wing and their effect on lift (2). These efforts have generally failed so that correlation with test data is poor. As in the case of low aspect ratio wings, the non-linear lift has been analyzed on basis of a drag based on the flow normal to the wing. This concept results in a lift increment equal to

CL = CDsin2cx cos a (3)

where Cq is the drag coefficient of the wing operating at an angle of attack equal to 90 . Fairly good correlation of test data is obtained with this procedure as indicated in conjunction with figure 7, Chapter XVII. However, at the lower angles of attack it appears that this is not the correct concept for calculating the change in lift due to the vortex flow of delta wings.

Leading-edge Suction Analogy. With the assumption of the leading-edge analogy (5,a, b,c, d) a procedure was developed that can be used to calculate the vortex lift increment. This method was developed for sharp edge delta type wings. The leading-edge suction analogy depends on the assumption that the thrust due to suction produced by the vortex in front of the wing is the same as that produced by the vortices on top of the wing. Thus, the vortices needed in the theoretical solution are not real in the sense of their location, since the experimental results show they are located on top of the wing, figure 3. By assuming that the thrust or suction force produced by the vortices is the same as produced by the actual vortices on the wing, lift increment is calculated by the equation

CL = sinz of cos of (4)

The value of Kv was developed in (5,a) for delta type, arrow and diamond planform wings with sharp edges and zero camber and is given in figure 10.

The value of К v for plain delta wings is approximately equal to Tf. This is considerably above any drag coefficient that might be expected for a delta wing operating at an angle of attack of 90°. It would, therefore, appear that the concept of drag is not valid for predicting the non-linear lift term of delta wings.

The calculated vortex and potential flow lift using equations 2 and 4 agree closely with measured values up to 20 as illustrated in figure 6. This correlation is typical for slender type flat delta wings with sharp edges which the theory assumes. Because of this excellent correlation a base is available for evaluating changes due to leading-edge radius, camber, flaps, control devices and bodies.

Vortex Lift to Total Lift. The vortex lift becomes the primary component as the sweep angle or slenderness of the wing is increased. This is illustrated by the data of figure 11 that shows the lift ratio as a function of aspect ratio.


Figure 10. Kv plot vortex-lift constant as a function of planform parameters.

Подпись: Figure 11. Comparison of vortex lift to total for delta wing A = 0 to 2.0.
Подпись: Figure 13. Lift characteristics of delta wings having sharp-edge (biconvex) airfoil sections tested by ARC (4) at low speeds.

Delta-Low Aspect Ratio Wing Analysis. The large difference shown in the value of in comparison to C q was not obtained with low aspect ratio wings, Chapter XVII. This could be due to the change in the potential lift curve between delta and low aspect ratio wings. The potential lift term determined for low aspect ratio wings is much higher than that for delta wings. Thus, the component due to the vortex flow is lower as determined from the experimental data for low aspect ratio wings. Since the potential flow can only be estimated from the test data, the question cannot be resolved. For this reason it is necessary to separate the analysis of the two types of wings until a more unified theory is developed.


Figure 12. Aerodynamic center of delta wings derived from experimental results.

The Aerodynamic Center. The chordwise position where the coefficient of the pitch moment is constant, aero­dynamic center, is theoretically:

at x/c = 0.25, in foil sections or straight wings

at x/jf =2/3, in slender delta wings.

Considering a delta wing to be a swept and highly tapered shape, a mean aerodynamic chord can be found using figure 2 of the first “longitudinal” chapter. As mentioned in connection with figure 1, this chord would be c = (2/3)X. In aspect ratios approaching zero, in very slender delta wings, the theoretical position of the aerodynamic center at x – (2/3) } , corresponds to 0.5 c, rather than to 0.25 c as in large A’ratios. However, we will refer the position of the center to the length j} of the delta wings considered,. Experimental results evaluated from dCm,/dCL slopes at small angles of attack (around zero lift) are plotted in figure 12. Assuming that the position, say at A = 10, would be close to x/j? = 0.5, it is seen moving toward the trailing edge as the A’ratio is reduced. All of the series of wings and plates investigated seem to approach the 2/3 position as their aspect ratio approaches zero. The limit up to which delta wings would

actually be used may be A = 2 (where $ /b = 1). In this planform shape, profiled (and round-edged) wings show positions х/л = 0.59. By comparison, flat plates have a more forward position, namely at 56% of the length (at A = 2).


Подпись: A = 4 tang, = 4 cot-Л Д/Ъ = о.5 tan Л = о.5 cots. Triangular wings with a straight trailing edge (point flying first) are called delta wings, according to their resem­blance with the Greek letter A . At least in larger aspect ratios they can be considered to be wings with zero taper ratio (pointed wing tips) and with a swept-back 4 chord line. Such wings would be impractical, however, because of wing-tip stalling and low lift drag ratios at subsonic speeds. Rather, delta wings are primarily designed for supersonic speeds. Their leading edges are highly swept so that the aspect ratio is usually small. Thus, the lift drag ratio at subsonic speeds is low. The leading edges of delta wings are also lateral edges. Circulation around such Ages is obviously different from what was originally meant by that word in two-dimensional flow past airfoil sec:ions; and the flow pattern is essentially different from that of straight wings, within the range of aspect ratios as used in airplanes.


In the range of low aspect ratio wings the delta shape is of particular interest. It is used in high speed aircraft, and there is analysis available explaining and predicting at least part of the lift.

Geometry. Wings in the shape of a delta (with a straight trailing edge) are obtained by combination of sweep and taper. As shown in figure 1, this type of wing is defined by the following parameters:

A = leading edge angle of sweep

€ = 90 – A = half vertex angle

A = 2 Ъ/Я ~ aspect ratio = b2/S

= 4/tanA. for delta wings.

If defining a mean aerodynamic chord in the manner as

Figure 1. Geometry of delta wings, illustrated by the example of an equilateral triangle.

shown for tapered wings in Chapter III, that chord measures c = (2/3)Я. However, since the load distribu­tion of slender delta wings is different from that in conventional planforms, there is no reason to use this type of mean chord. In straight delta wings the length “ У ” would be a natural dimension to define a length ratio. However, to accommodate shapes differing in contour (such as parabolic or “gothic” planforms) the mean geometric chord c = S/b == £ /2 shall be used.

When the trailing edge of delta wings is modified from a straight line the planform is changed to an arrow or a diamond shape as is illustrated on figure 2. For an arrow wing, the wing notch ratio a/f is positive while for the diamond shape a/£ is negative. The notch parameter is used to determine the coefficients for finding the vortex and potential flow lift.