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Characteristics of Sweptback Wings
Qualitatively, the effect of wing sweep can be seen by referring to Figure 5.26. In Figure 5.26a, a wing section is shown extending from one wall of a wind tunnel to the other. The test section velocity is denoted by V„. Imagine that the wing in Figure 5.26a is only a section of an infinitely long wing that is being drawn through contoured slots in the tunnel walls at a velocity of v. Obviously, the pressure distribution around the section does not depend on v. Vn and v combine vectorially, as shown in Figure 5.26b, to give a velocity of У» relative to the wing. As shown in Figure 5.26c, this is equivalent to a swept
Figure 5.26 Effect of sweepback.
wing of an infinite aspect ratio with a sweep angle of Л and a free-stream velocity of V„ Thus, the chordwise Cp distribution of such a wing depends only on the component of normal to the span, which is given by
V„ = V. cos ft (5.82)
Based on this velocity, a pressure coefficient can be defined as
The Prandtl-Glauert transformation can then be applied to CP„ using M„ in order to account for compressibility. In practice, a swept wing has a finite length. Near the apex of the wing and at the tips, a three-dimensional flow effect will be encountered. Indeed, one cannot test a two-dimensional swept wing in a wind tunnel (except in the manner shown in Figure 5.26a). For example, a wing placed wall to wall and yawed in a wind tunnel models a saw-toothed planform instead of an infinitely long swept wing. As illustrated in Figure 5.27, this results from the fact that the flow must be parallel to the wind tunnel walls at the walls. This can only be satisfied by assuming an image system of wings having alternating sweep, as shown.
The effect of sweepback on critical Mach number can be estimated using Equation 5.83. For example, suppose a straight wing has a certain chordwise and spanwise Cp distribution that produces a given lift. At some point on the wing, suppose that a minimum C„ is equal to -0.5. According to Figure 5.19, its Mcr value would equal 0.71. Now suppose the same wing were swept back 45° and its twist, camber, and angle of attack were adjusted to give the same chordwise Cp distribution (based on VJ) at each spanwise station as for the unswept wing. The total lift for the two wings would then be the same. For the swept wing the minimum Cp based on V„ becomes CPn = -1.0. Thus, according to Figure 5.19, M„cr = 0.605. This corresponds to a free-stream
critical Mach number, M„cr, of 0.86. Therefore, in this case, sweeping the wing back 45° for the same wing area and lift has increased the critical Mach number from 0.71 to 0.86.
Sweepback is also beneficial in supersonic flow. Mach waves propagate from the leading edge of the wing at the angle, p (Equation 5.45). If the sweepback angle, Л, is greater than the complement of p, the flow component normal to the span is subsonic. Locally the resultant flow along a streamline is still supersonic, but the Mach waves generated at the leading edge deflect the flow, thereby lessening the strength of the oblique shocks.
Sweepback is not without its disadvantages, so it is normally used only when called for by compressibility considerations. Sweeping a wing will cause the loading to increase toward the tips unless it is compensated for by washout. At the same time, the spanwise component of produces a thickening of the boundary layer in the tip region. Hence a swept wing is more likely to stall outboard by comparison to a straight wing; this characteristic is undesirable from the standpoint of lateral control. Also, tip stall (which might occur during a high-speed pull-up) can cause a nose-up pitching moment, further aggravating the stall.
Aeroelastic effects caused by sweep can also be undesirable. A sudden increase in angle of attack can cause the wing to bend upward. As it does, because of the sweep, the tips tend to twist more nose downward relative to the rest of the wing. Again, this can produce a nose-up pitching moment that increases the angle of attack even further. This behavior is an unstable one that can lead to excessive loads being imposed on the airframe.
Let us refer once again to Figure 5.26c. A rotation of a about a line along the wing is shown as a vector in the figure. Observe that the component of this vector normal to V» is equal to a cos Л. Thus an angle of attack of a relative to Vn results in a smaller angle of attack of a cos П relative to V.
The lift on a unit area of the wing will be given by
L = hpV2C, a = 5p(V„cos Л )2C, a
where Cia is the slope of the lift curve for an unswept two-dimensional airfoil section. The corresponding quantity for a swept section can be obtained by dividing the preceding equation by the free-stream dynamic pressure and the angle of attack relative to V„.
C, JA Ф 0) = Ll(pVja cos Л)
= Ci cos Л (5.84)
*a
Reference 3.35 presents an approximate equation for the lift curve slope of a wing with sweepback in a subsonic compressible flow. The equation is derived by assuming that Equation 3.74b holds for swept wings using the
section lift curve slope for swept wings. Compressibility is accounted for by applying the Prandtl-Glauert correction to Equation 5.84 using Repeating Equation 3.74b for an unswept elliptic wing,
(а/тт) + V(a/-77-)2 + A2 where a denotes Cta. Substituting
a = a0 cos Лі/2/V 1 – Mj cos2 Л1/2 the expression for CLa reduces to
(a0lir) + V(A/cos A,/2)2+- (а0/тг)2 – (AM*)2
where a0 = Cta for Л,/2 = M = 0. Note that a subscript і has been added to Л to indicate that Лi/2 should be measured relative to a line through the midchord points. It is argued in Reference 3.35 that the use of A1/2 makes Equation 5.85 independent of taper ratio, A. This conclusion appears to be supported by Figure 5.28 (taken from Ref. 3.35). Here, Equation 5.85 (for M = 0) is seen to
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Lifting surface calculations X Af/? cos Лу2 Figure 5.28 CLJA with Д/cos Л1/2 as determined by several methods. a0 = 2n, M =0. |
compare favorably with several lifting surface calculations for elliptical and tapered wings.
A tabulation of subsonic lift and moment characteristics for tapered, sweptback wings is presented in Table 5.2. Taken from Reference 5.12, these results are based on a numerical lifting surface theory. They should be more accurate than Equation 5.85, although not as convenient to use. As an example in the use of this table, let
/3 A = 5.0 /3 =0.6 A tan Л ]/2 = 4
It then follows that
A = 8.333 Am = 25.6° f3CLa = 3.43 CLa = 5.717
For this example, Equation 5.85 gives a Cia value of 6.017, which is about 5% higher than the lifting surface theory result. Of course, both results can be expected to be somewhat high by comparison to experiment, since the theoretical value for a0 of 27rC(/rad is a few percent higher than that found experimentally for most airfoils.
Figure 5.29 (from Ref. 5.14) presents some experimental results on CLa at low Mach numbers. Generally, the trends shown on this figure confirm the predictions of Figure 5.28. The two figures are not directly comparable, since the sweep angles of Figure 5.29 are relative to the leading edge instead of to midchord. To go from one reference angle to the other requires a knowledge of the taper ratio as well as the aspect ratio.
Figure 5.30 (taken from Ref. 5.13) presents a limited amount of information on Cl^ for swept wings with various combinations of flaps. Remembering the “independence principle” for the velocity normal to the sweep line, one might expect that would decrease as cos2 ft. However, referring to the data points in Figure 5.30 without flaps, this variation with cos2 Л does not appear to be valid. For example, for Л = 45°, cos2 Л = yet the С^ах values for these swept wings are certainly greater than half of what one would expect for the unswept wings. Three-dimensional effects such as spanwise flow and leading edge vortices undoubtedly play an important role in determining the stalling characteristics of a swept wing. Figure 5.31 indicates that, if anything, there is a tendency for C^ to increase with sweepback (or with sweep forward). Admittedly, this figure includes other factors affecting C/max, but the general impression that it portrays is probably valid; that is, sweep has little effect on С^ах.
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Table 5.2 Subsonic Theoretical Lift Slopes, Pitching Moments, and Aerodynamic Centers for Wings of Varying Sweep and Taper (M about leading edge at midspan, C = geometric mean chord)
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Table 5.2 (continued)
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The effect of sweepback on drag can be quickly summarized by Figures 5.32 and 5.33. Figure 5.32 allows one to estimate the induced drag as a function of sweepback, aspect ratio, and taper ratio. These graphs were prepared using Weissinger’s first-order lifting surface theory. The bound vortex is placed at the quarter-chord points with control points at the three-quarter-chord points. (See the equivalent two-dimensional calculation leading to Equation 3.19.) These graphs apply only to = 0, so they must be used in conjunction with the Prandtl-GIauert three-dimensional transformation. Qualitatively the graphs are reasonable. A rectangular wing (A = 1) is already loaded more heavily toward the tip, so that sweep simply aggravates ‘ the situation, causing a continuous departure from the ideal elliptic loading. For the strongly tapered wing (A = 0) the situation is reversed so that, in this case, sweep is beneficial. For the taper ratio of 0.25 there is practically no effect of sweep on the induced drag.
Figure 5.33 indicates that there is little, if any, effect of sweepback on the minimum drag coefficient. Thus one can estimate CD on the basis of
° ‘-‘min
two-dimensional airfoil measurements. Usually these drag measurements are taken at low Mach numbers so that it is necessary to correct the skin friction part of Cn. for compressibility effects. Such a correction is given by the graph of Figure 5.34 (Ref. 5.1) for Mach numbers as high as 10.0.
An empirical equation that closely fits the graph of Figure 5.34 is
0.04 0.0?
T I 1 Г I c I I
0 10 20 30 40 50 60
лс/4, deg
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Figure 5.33 Variation of the minimum drag coefficient with sweep angle for a family of wings having aspect ratios of 4, taper ratios of 0.6, and NACA 65A006 airfoil sections parallel to the plane of symmetry. |
Mach number, M
Figure 5.34 Variation of skin friction coefficient with Mach number for a turbulent boundary layer at zero heat transfer. Cfi is the value for incompressible flow. Solid and dashed lines represent two different theoretical solutions. (H. W. Liepmann and A. Roshko, Elements of Gas Dynamics, John Wiley & Sons, Inc., 1957. Courtesy of U. S. Naval Ordnance Laboratory.)
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tapered swept wing, ALe and Л|/2 can easily be shown to be related by
Thus, for this case, A1/2 = 25.7°. The incompressible slope of the lift curve, a0, for this particular airfoil equals approximately 6.2 CJrad. Thus, for = 0.7, using Equation 5.85, CLa = 5.21 Сь/rad. |
According to Equation 5.66, the equivalent wing in incompressable flow will have its у dimensions decreased by the factor /3. In this case, /3 = 0.714. Hence the equivalent wing’s geometry becomes
Ai = 5.0
Л,1/4= 36.6°
For this equivalent wing, from Figure 5.32,
For the given wing loading and operating conditions,
WIS
= 0.462
1 (0.462)2 0.98 77-(7)
= 0.00990
We do not need to correct the profile drag for compressibility effects in this example, since Figure 5.3 provides us with the section Cd as a function of M and С/. Assuming that the section C( is equal approximately to the wing CL of 0.462, Cd is read from Figure 5.3 to be 0.015 for an of 0.7. According to Figure 5.33, this does not need to be corrected for sweep. The total wing CD at the operating CL and Mach number is thus estimated to equal 0.0249.
Before leaving this example, it is of interest to generalize on the calculation of Сц.
Ai = (3AC Cl, = (32CLc
Thus,
= /3 3CDi (5.90)
‘c
This result can also be obtained by applying the three-dimensional Prandtl – Glauert transformation to the relationship
Сц — C;a,
ati, the induced angle of attack, is proportional to the vertical velocity associated with ф(х, fiy, /3z). Hence, a, varies as 1//3.
Delta Wings
The geometry of a delta wing is pictured in Figure 5.35a. Such a wing is also referred to as a triangular wing. The delta wing represents the limiting
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у |
Figure 5.35a The planform of a delta wing.
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Figure 5.35b Two-dimensional flow approximation for a slender wing. |
The aerodynamic analysis of a delta wing in the general case is not an easy task because of nonlinear effects associated with the highly swept leading edge. At small angles of attack and low aspect ratios one can apply a linearized approach that is referred to as the slender wing (or wing-body) theory. The details of this theory will not be developed here; instead, refer to two of the original sources (Refs. 5.17 and 5.18) for the treatment of this theory. A brief treatment for the wing-alone case can be found in Reference
3.3. Basically, the theory assumes that the flow in any transverse plane is essentially two dimensional. This assumption eliminates the Mach number- dependent term in Equation 5.56 that governs the perturbation velocity potential, leaving only
(5.93)
Since V„a is the velocity component normal to the plane of the wing, the problem is reduced to finding the two-dimensional flow in a transverse plane, as illustrated in Figure 5.35b.
For a flat wing, the pressure difference across the wing is predicted to
where bx is the local span a distance of jc from the apex, y, bx, and x are shown in Figure 5.35b.
Given dbjdx as a function of x, Equation 5.94 can be integrated over the surface of the wing to obtain the wing lift coefficient. For a delta wing, dbjdx
is constant, giving an elliptical spanwise loading. In this case,
(5.95 a)
(5.95 b)
Notice that because of the assumption of slenderness, these results do not depend on Mach number.
Since the shape of the local spanwise pressure distribution is the same at all chordwise positions for a delta wing (dbjdx is constant), it follows that the center of pressure is predicted to be at the centroid of the planform area, a distance of 2cJ3 back from the apex. Based on the wing area and geometric mean chord, the pitching moment coefficient about the apex is thus
(5.96)
Again, these results hold for any Mach number, providing the aspect ratio and angle of attack are sufficiently small. Depending on the desired accuracy, the limitations on A and a can be severe, possibly as low as 0.5 for A and 2 or 3° on a.
In order to predict the behavior of delta wings for higher a’s and aspect ratios, it is necessary to rely on physical observations of the flow over such wings to form the basis for an analytical model. In subsonic flow a delta wing is observed to generate a vortex located just inboard and slightly above each leading edge.
A striking photograph showing the formation of these vortices is presented in Figure 5.36a. This photograph was taken in a water tunnel with dyed milk ejected from holes in the model wing in order to illustrate the streamlines. To clarify the flow further, the generation of these vortices is shown in Figure 5.37a. Figure 5.37b depicts the flow in a plane more or less normal to the leading edge. Observe that as the air flows from beneath the wing around the leading edge, it is unable to remain attached and separates from the upper surface. It spirals around and reattaches to the upper surface some small ^distance inboard of the leading edge.
According to Reference 5.19, this separated standing vortex is equivalent to the separated flow region at the leading edge of a thin airfoil, as shown in Figure 5.37a. The ideal potential flow case of a fully wetted, thin airfoil is shown in Figure 5.37b, where a singularity exists at the leading edge (see Equation 3.36). In view of D’Alembert’s paradox, we know that a drag cannot exist in the case of Figure 5.37b. Therefore, the infinitely negative pressure acting over the vanishingly small leading edge, in the limit, must produce a force to cancel exactly the component in the drag direction of the normal pressure difference integrated over the chord. This force is known as the
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Figure 5.36a The vortex system of a delta wing. (Courtesy of the Office National D’Etudes Et De Recherches Aerospatiales.) |
leading edge suction force. Its resultant lies in the plane of the chord and, hence, has components in the lift and drag directions.
The leading edge suction force is essentially independent of the leading edge radius. Thus, if the separated streamline in Figure 5.37a is replaced by a solid boundary, a leading edge suction force will be present. This suggests that without the boundary, the force on the separated flow region from the reduced pressure resulting from the curved flow must be approximately equal to the leading edge suction force. Assuming a circular flow (which is not quite true), this leads to a reaction in the upward direction normal to the chord and near the leading edge that is nearly equal in magnitude to the leading edge suction force for the fully wetted, potential flow case. This is the basis for the subsonic Polhamus delta wing theory presented in References 5.19 and 5.20.
Without delving further into the details of Polhamus’s theory, the following results are obtained.
CL = Kp sin a cos2 a + Kv cos a sin2 a (5.97)
The term Kp sin a cos2 a represents the potential lift on the wing for the fully wetted case minus a small vertical component of the leading edge suction force, which is lost when the flow separates. The second term represents the added lift caused by the presence of the leading edge vortex. Not surprisingly, this is referred to as “vortex lift.”
As a approaches zero, Equation 5.97 becomes
CL = Kpa (a -» 0)
Thus it follows that the constant, Kp, is simply the slope of the wing lift curve for the fully wetted case according to the usual lifting surface theories. The constant, Kv, is obtained from
К = Kp~ KeKi (5.98)
” cos Ale
where Kj = dCoJdCb and ALe is the sweepback angle of the leading edge.
Again, Kj can be evaluated from the usual lifting surface methods. However, Kp and Kv have already been evaluated in the references and are presented here in Figure 5.38 as a function of aspect ratio.
Having CL, the drag coefficient is easily obtained. Since there is no leading edge suction force parallel to the chord for the separated leading edge, it follows simply that
CD = CL tan a (5.99)
Equations 5.97 to 5.99 hold only for M = 0. The effects of compressibility on these equations can be estimated by again using the Prandtl-Glauert transformation. Given an aspect of A, one calculates Kp (designated K’p) for a
transformed aspect ratio A’ given by,
A’ = PA
Kp for the original wing is then found from
—- Cl (Equation 5.97)
——Cl No vortex lift
Figure 5.39a Comparison of theory with experimental data for sharp-edged delta wings.
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Figure 5.39b Comparison of experimental drag due to lift with the results of three theoretical methods (M – 0). |
for the loss of leading edge suction. Figure 5.39h also includes predictions based on fully wetted potential flow. Obviously, the potential flow results are questionable, whereas the predictions based on the leading edge suction force analogy agree closely with experiments up to angles of attack as high as 25°.
An estimate of for thin delta wings at low speeds can be gained 4 from Figure 5.40. These points represent a miscellaneous collection of wings with different sweepback angles and thickness ratios. Unfortunately, systematic data could not be found to correlate C;^ax with all of the pertinent parameters. Based on this limited presentation of data, it appears that aspect ratios between 1.5 and 2.5 can be expected to deliver values of at least
1.3. From Figure 5.39a, it can be seen that a CL value of this magnitude corresponds to an extremely high angle of attack greater than 25°. In order to provide adequate forward vision during a landing, because of the high angle of attack, the Concorde supersonic transport pictured in Figure 5.25 droops the nose of the fuselage.
