AN ALTERNATIVE APPROACH TO INDUCED DRAG REDUCTION

. Induced drag constitutes approximately 40% of the total drag of subsonic civil transport aircraft at cruise conditions. Various types of winglets and several non-planar concepts, such as the C-wing, the joined wings, and the box plane, have been proposed for its reduction. Here, a new approach to induced drag reduction in the form of a combination of an elliptical and an astroid hypocycloid lift distribution is put forward. Lift is mainly generated from high circulation in the center part of the wing and fades away along the semi-span towards the wing tip. Using lifting line theory, the analysis shows that for fixed lift and wingspan the combined lift distribution results in an induced drag reduction of 50% with respect to the elliptical distribution. Due to its wing planform the combined lift distribution leads to a 51.5% higher aspect ratio. If structural constraints are placed, then the higher aspect ratio may affect wing weight. Although any substantial increase of wing weight is not envisaged, further study of the matter is required. Zero-lift drag and lift-dependent drag due to skin friction and viscosity-related pressure remain unaffected. The proposed lift distribution is particularly useful in a blended wing-body design.


Introduction
Fuel costs and greater awareness of the impact of emissions on the atmosphere raise the importance of fuel efficiency for future transport aircraft. Fuel consumption can be reduced by decreasing airframe weight or drag and improving the efficiency of the propulsion system (Allison et al., 2010). At subsonic speeds, aircraft drag is caused by two basic phenomena: the influence of viscosity, primary through skin friction, and losses associated with the generation of wing lift (Kroo, 2001). Induced drag is liftdepended drag as the result of trailing wing tip vortices. Induced drag accounts for nearly 40% of the total drag of subsonic civil transport aircraft at cruise conditions (Howe, 2000;Torenbeek, 2013;Kroo, 2001).
Reduced induced drag is achieved by increasing aspect ratio and/or span efficiency factor. Increasing aspect ratio decreases wing vortex strength, and hence it results in lower induced drag. The disadvantages of this approach include increased wing structural weight due to higher wing bending moments. The other option is to reduce induced drag by increasing wingspan efficiency factor. Towards this end several concepts and schemes, such as winglets (Whitcomb, 1976;Chattot, 2006;Demasi et al., 2019) and non-planar lifting surfaces in the form of the C-wing (McMasters & Kroo, 1998), the joined wings (Wolkovich, 1985) and the box wing (Prandtl, 1924;Scardaoni, 2020), have been proposed. Among them, only winglets have been used in operational aircraft. The reason is that non-planar concepts are complex and the source of problems in the overall design of aircraft (Kroo, 2001). Winglets have an induced drag reduction potential of 20 to 25% (Whitcomb, 1976;Demasi et al., 2019).
Lift distributions of plain planar wings for fixed lift and wingspan without any further constraints generating less *Corresponding author. E-mail: n.kehayas@gmail.com induced drag than Prandtl's (1921) elliptical distribution are not known.
In the present study a new approach in the shape of a combination of an elliptical and an astroid hypocycloid lift distribution is put forward with the objective to reduce induced drag. The motivation is to find a way to decrease induced drag of plain planar wings simpler and more effective than existing non-planar concepts or winglet designs. The elliptical lift distribution spans the center part and the astroid hypocycloid the outer parts of the wing. This combined distribution reduces wing vortices by smoothly fading away lift towards the wing tip. Although most of the lift is provided by the elliptical distribution in the center part of the wing, the astroid hypocycloid distribution in the outer parts of the wing contributes to lift as well. Following lifting line theory, the induced drag of the proposed combined lift distribution is evaluated, and then compared with the induced drag of the elliptical lift distribution, nonplanar concepts, and winglet designs. The proposed lift distribution is examined as to its application to aircraft design.

Analysis
It has been shown by Prandtl's lifting line theory that the lift distribution which yields the minimum induced drag for fixed lift and wingspan is the elliptical lift distribution (Anderson, 1984). This is the minimum vortex drag and does not include lift-dependent drag due to skin friction and viscosity-related pressure drag (Kroo, 2001). The induced drag coefficient of wings with an elliptical lift distribution is given by The proposed combined lift distribution is shown in Figure 1. The two lift distributions blend at the point where they have the same gradient and value of circulation. Further down in the analysis this point is found to be 0.71(b e /2), 0.71(b a /2), where b e /2 and b a /2 are respectively the elliptical and the astroid hypocycloid distributions semi-spans. The elliptical part of the combined lift distribution spans from the origin of the semi-span of the combined lift distribution, 0 c, to the blending point of the two distributions, 0.71(b e /2), 0.71(b a /2), and the astroid hypocycloid part spans from the blending point, 0.71(b e /2), 0.71(b a /2), to the end of the semi-span of the combined distribution, b/2.
The following analysis applies to vortex drag of unswept, untwisted plain planar wings for fixed lift and wingspan without any other constraints in incompressible flow.
For an elliptical lift distribution (Anderson, 1984) along the span of the wing, the circulation distribution, Γ e (y), is given by where y is the distance along the wing semi-span and b e /2 is the semi-span of the wing based on an elliptical lift distribution.
For an astroid hypocycloid (Simmons, 1992) lift distribution along the span of the wing, the circulation distribution, Γ a (y), is given by where y is the distance along the wing semi-span and b a /2 is the semi-span of the wing based on an astroid hypocycloid lift distribution.
As already mentioned and shown in Figure 1, the elliptical lift distribution covers the center part and the astroid hypocycloid the outer parts of the wing. The two lift distributions blend at the point where they have the same gradient and value of circulation. For the same circulation gradient For an elliptical circulation distribution with a minor to major semi-axis ratio of 1/2 we have and for an astroid hypocycloid circulation distribution with an axial ratio of 1 we have ( ) Use of equations (2) and (3) to evaluate equation (4) and then substituting equations (5), (6) and (7) into it gives whose solution is 0.71 Y = ± . The two distributions do not have the same origin in the y (wingspan) axis and the blending point is located at For the same circulation value at the blending point .
Hence, from equations (5), (6) and (11) From the above, as indicated in Figure 1, the semi-span of the combination of the two lift distributions, b/2, is and using equation (13) 1.818 2 2 According to Anderson (1984) induced velocity, w(y o ) , which is the cause of induced drag, is given by is the induced velocity due to the elliptical lift distribution of the central part of the wing, and are the induced velocities due to the astroid hypocycloid lift distribution of the outer parts of the wing.
To continue with the evaluation of the integrals for the induced velocity and lift, a transformation of variables is needed (Anderson, 1984). It is the transformation of the linear variable of the distance along the semi-span, y, into an angle variable, θ, with respect to the corresponding lift distributions semi-spans, b e /2 and b a /2. The reason is to facilitate the evaluation of these integrals, especially the induced velocity integral of the elliptic lift distribution.
For the induced velocity, w e , due to the elliptical part of the lift distribution, by using equation (2), we have Considering the transformation of the distance along the semi-span variable, y, into an angle variable, θ, for an elliptical distribution, the following substitution is applied (For clarity purposes angle limits are rounded to the second decimal place, resulting in an error of less than 1%). Hence, To evaluate induced velocity, equation (23), use of the Glauert Integral (GI) (Glauert, 1947) is made. Therefore, and hence, For the induced velocity, w a , due to the astroid hypocycloid parts of the lift distribution, using equation (3), we have Considering the transformation of the distance along the semi-span variable, y, into an angle variable, θ, for an astroid hypocycloid distribution, the following substitution is applied (For clarity purposes angle limits are rounded to the second decimal place, resulting in an error of less than 1%). Consequently, Or, using equations (11) Similarly, Hence, Considering the transformation of the distance along the semi-span variable, y, into an angle variable, θ, for an elliptical distribution, the following substitution is applied and hence, For the lift, L a , due to the astroid hypocycloid part of the distribution we have Considering the transformation of the distance along the semi-span variable, y, into an angle variable, θ, for an astroid hypocycloid distribution, the following substitution is applied According to Simmons (1992) Next, in order to find the downwash, w, and the lift, L, of the combined elliptical and astroid hypocycloid lift distribution, the downwash, w, and the lift, L, for each distribution must be reduced to the span, b, of the combined distribution.
Using the definition of induced velocity equations (15) and (25) and using the definition of induced velocity equations (15) and (39) Hence, the downwash of the combined distribution is Using equations (15) and (43) Further, according to Anderson (1984) the induced angle, a i , is given by and using equation (53) ( ) From equation (58) ( ) Hence, Substituting equation (63) into equation (60) gives According to Anderson (1984) the induced drag coefficient, C di , follows from induced drag, D i , given by Using equation (64) i i Substituting equations (63) and (64) To be possible to compare the induced drag found for the combined elliptical and astroid hypocycloid lift distribution (equation (72)) with the induced drag of the elliptical lift distribution for the same lift and wingspan, the above result (equation (72)) must be further examined.
Keeping all other parameters constant, the wing surface (planform), and consequently the lift of the combined elliptical and astroid hypocycloid lift distribution is 66% of the elliptical (equation (58)). To compare the induced drag of the combined lift distribution with the elliptical for the same lift and wingspan, the lift of the combined distribution must be brought up to the level of the lift of the elliptical. For the same wingspan, the combined lift distribution has a wing surface which is 66% that of the elliptical. Consequently, for the same lift and wingspan the lift coefficient (circulation) of the combined, C lc , must be increased by 51.5 % with respect to the elliptical, C le (1.515 × 0.66 = 1.00).
Using the subscripts c for the combined and e for the elliptical

Results and discussion
As shown in the analysis the use of a combined elliptical and astroid hypocycloid lift distribution results in an induced drag coefficient reduction of 50% (equation (70)). This reduction translates into a span efficiency factor, e, of 2. At this value the combined lift distribution outperforms all non-planar concepts (Kroo, 2001). In most cases the effect is more pronounced if total drag and lift-to-drag ratio are considered. In addition, the proposed combined lift distribution is free of the design problems experienced by non-planar aircraft configurations. The induced drag coefficient reduction of the combined elliptical and astroid hypocycloid lift distribution is two and a half times the reduction achieved by winglets (Whitcomb, 1976). The most elaborate winglet designs, such as the box winglets (Demasi et al., 2019), exhibit span efficiency factors of less than 1.5. Again, as with most non-planar concepts, the reduction is even greater if total drag and lift-to-drag ratio are considered. Winglets reduce induced drag, but they produce little lift. In contrast, the astroid hypocycloid part of the combined lift distribution generates nearly one-third of the total lift but only one-sixth of the total downwash (Table 1). That is, in relation to the elliptical part it produces with respect to lift proportionally less downwash.
Although with the lift-to-downwash ratio as the criterion the astroid hypocycloid is more efficient than the elliptical part of the combined distribution, both, as indicated in Table 1, are more efficient than the elliptical. Not only combined, but "separately" as well they are more efficient in relation to lift-to-downwash ratio. And this effect is not due to their respective distributions but due to the section of each distribution used and its wingspan location.
The comparison of the proposed combined lift distribution with non-planar concepts and winglet designs is based on an overall height to span ratio of 0.2. For values of this ratio beyond 0.5, box plane, biplane and winglet concept performance is comparable to the combined lift distribution in terms of span efficiency factor (Kroo, 2001). However, this is only a theoretical consideration because at these overall height to span ratios these concepts exhibit serious drawbacks in relation to profile drag, structural weight, stability, and other aircraft design issues (Torenbeek, 2013).
For fixed lift and wingspan, the use of a combined elliptical and astroid hypocycloid lift distribution results in an induced drag reduction of 50% compared with the elliptical (equation (79)). The physical interpretation of this reduction is that lift is mainly produced from high circulation in the center part of the wing and fades away along the semi-span towards the wing tip. Therefore, the vortices produced by the wing are weaker. It is the outcome of replacing the elliptical lift distribution in the outer parts of the wing with the more efficient lift-to-downwash ratio astroid hypocycloid.
Fixed lift and wingspan is the established way to compare different lift distributions. Prandtl (1921) in his lifting line theory demonstrated that the lift distribution which yields the minimum induced drag for fixed lift and wingspan is the elliptical. Prandtl (1921, p. 191), states that various lift distributions were attempted, and that the elliptical gave the desired solution. He does not indicate a mathematical process towards this solution. Munk (1923, p. 7), using calculus of variations arrived at the conclusion that the necessary condition for the minimum of the in-duced drag is that the downwash produced to be constant along the entire lifting line.
Although the result reached in the analysis is rather unconventional, nowhere, at least not in Prandtl's (1921) and Munk's (1923) work, there is a method for deducing that the elliptical lift distribution minimizes induced drag. If a "trial and error" approach was used by Prandtl (1921) to discover the elliptical as the distribution of least induced drag, it is highly unlikely that every form of lift distribution or combination of lift distributions was investigated. Munk's (1923) finding that the downwash must be constant does not define the form of the lift distribution. The fact that the elliptical lift distribution produces a constant downwash confirms that it is the distribution producing the minimum of induced drag and not the other way around. Equally, the general lift distribution expressed as a Fourier sine series (Prandtl, 1921) implies with its first term by definition an elliptical distribution.
In the analysis by means of a numerical integration a constant downwash is reached for the astroid hypocycloid part of the combined lift distribution (equation (37)). Downwash is estimated in a numerical manner due to the great difficulty in evaluating the integral of the downwash for the astroid hypocycloid part (equation (32)). But the downwash gradient of the astroid hypocycloid part, -dw/ dθ, can be very closely described in the domain of its values [0.15π -0] by the function whose integration from 0.15π to 0 gives for the astroid hypocycloid part downwash the same value analytically as numerically. That is Consequently, it is shown analytically, albeit in an indirect way, that downwash is independent of the position at which it is induced, and thus it is constant over the domain [0.15π -0] of the astroid hypocycloid part. Hence, both elliptical and astroid hypocycloid parts of the combined lift distribution fulfill the condition of constant downwash in their respective domains of wingspan. Therefore, it may be claimed that the use of a combination For fixed lift and wingspan, the combined elliptical and astroid hypocycloid lift distribution results in an induced drag reduction of 50% in comparison with the elliptical. If structural considerations like wing root bending moment are included in the comparison, then a different outcome arises. This is not the outcome reached by Prandtl (1933) or of any similar investigations involving structural constraints. The reason is that the planform of the combined elliptical and astroid hypocycloid lift distribution, with its large center wing area section due to its elliptical part and its small outer wing area sections due to its astroid hypocycloid part (a reflection of Figure 1), is unlike any other. This planform gives rise to a 51.5% higher aspect ratio in comparison with the elliptical. But structural effects due to a higher aspect ratio are not as adverse as they seem since the small outer wing area sections generate only one-third of the total lift (Table 1). If structural or other constraints are placed, such as bending moments, wing weight, maximum stress, wing loading and wing deflection (Phillips et al., 2019), further study of the proposed combined lift distribution is required which is beyond the scope of this paper.
The above results and discussion refer to vortex drag of plain planar wings without twist or sweep. Nevertheless, induced drag includes, even though very much smaller than vortex drag, a lift-dependent drag due to skin friction and viscosity-related pressure (Kroo, 2001). This term is related to airfoil shape and not to wing lift distribution. In this sense it is irrelevant to this analysis. But it should be noted that because of the considerable vortex-dependent drag reduction of the combined lift distribution, liftdependent drag due to skin friction and viscosity-related pressure may now be almost in the same order of magnitude as vortex-dependent drag.
The proposed combined lift distribution could be of much use in a blended wing-body design (Kehayas, 1998). The elliptical part of the combined lift distribution could be applied to the passenger-carrying large wing area center section, and the astroid hypocycloid to the small outer wing sections of the blended wing-body. Use of lift and propulsion integration schemes (Kehayas, 2006(Kehayas, , 2011 would more than alleviate any wing weight penalties resulting from the higher aspect ratio of the combined lift distribution.

Conclusions
The main conclusion is that the proposed combination of an elliptical and an astroid hypocycloid lift distribution for fixed lift and wingspan planar wings without any other constraints leads to an induced drag reduction of 50% in comparison with the elliptical. The physical interpretation of this reduction is that lift is mainly produced from high circulation in the center part of the wing and fades away along the semi-span towards the wing tip reducing wing vortices. Compared to the elliptical part of the combina-tion the astroid hypocycloid generates with respect to lift proportionally less downwash. The induced drag reduction of the combined lift distribution is not due to the elliptic and the astroid hypocycloid distributions it is made from, but due to the section of each distribution used and to its wingspan location. The proposed lift distribution outperforms all non-planar concepts and winglet designs with a feasible overall height to span ratio. The wing planform of the combined lift distribution exhibits a 51.5% higher aspect ratio. If structural constraints are placed, then the higher aspect ratio may affect wing weight. Although any substantial increase in wing weight is not envisaged, further study of the matter is required. Zero-lift drag and liftdependent drag due to skin friction and viscosity-related pressure remain unaffected. The proposed lift distribution is particularly useful in a blended wing-body design.

Disclosure statement
The author declares not to have any competing financial, professional, or personal interests from other parties.