## 1. Introduction

The potential advantage of using non-planar lifting surfaces to reduce induced drag has long been known. These include closed wings, C-wings and wings with winglets. Despite recent technological advances and the use of high rigidity Carbon Fiber Reinforced Plastic wing panels that allow very large aspect ratios, which greatly cut the induced drag at cruise, most airframe manufacturers are still applying winglets, both for further pushing the boundaries of the wing aerodynamic efficiency and due to their aesthetic value. Since their introduction in 1975 by National Aeronautics and Space Agency (NASA) engineer Richard Whitcomb [

1,

2], winglets have undergone major geometrical changes, driven by the quest for a sustainable efficiency throughout the entire flight envelope of modern airliners. Due to the largely different flow field nature at the wing tip area at different flight conditions, and the challenge of achieving a winglet shape optimal throughout the whole flight envelope, recent winglet geometry optimization studies in the last decade were mostly focused on cruise as a ‘design case’ for this wingtip device, resulting in trade-off solutions less optimal for take-off and climb at high angles of attack [

3,

4,

5,

6,

7,

8,

9,

10]. Few innovative solutions have been suggested including morphing winglets which adapt their cant and/or twist depending on the flow regime [

11,

12,

13], or using an integrated moving device such as a winglet-integrated rudder [

14,

15] or a gust alleviating conventional aileron [

16], as well as active vortex wake control with an oscillating mechanism [

14,

17,

18]. Bio-inspired devices were studied with a triangular leading-edge extension [

19], with multiple elements, where each element is set at a given angle and contributes at a certain flight regime, thus sustaining overall efficiency [

20,

21]. To date, none of these novel solutions could find application on an operational airliner, mainly due to their high complexity which both compromises reliability, and involves a weight penalty, potentially cancelling their aerodynamic gains.

Most winglet geometry optimization attempts [

3,

4,

5,

6,

7,

8,

9,

10] are purely experimental, based on the results of either computational and/or wind tunnel experiments on a large population of samples with a subsequent optimization using a statistical approach such as pareto front. Unfortunately, the general lack of a winglet local-flow-field theory makes it challenging to explain experimental results, and to uncover proper improvement directions for future designs. Keizo et al. [

3] performed high fidelity CFD and wind tunnel multidisciplinary design exploration of a commercial jet winglet, which provides a fairly deep insight into the winglet design space, allowing an optimal design decision based on pareto front and simple trade studies. Although this is a pragmatic approach usually used by airframers to retrofit operational airliners with winglets [

6,

7], it results in an improved initial geometry, rather than a novel design, due to the lack of a closed-form relationship between the winglet geometry and overall lift-to-drag ratio. Given the winglet is a finite span lifting surface, classical wing theory has been extended to predict the winglet performance. For instance, a CFD combined with the lifting line theory approach has been used by Jan Himisch [

4] and Streit et al. [

6] to obtain an optimal cruise

L/D ratio for a generic transonic airliner. An attempt of finding a winglet optimal twist was performed by Neal [

10], by relating the winglet root and tip incidence angles with the induced drag reduction, obtained as output from potential flow solutions in the Trefftz plane behind the wing. However, there are virtually no recent theoretical or experimental studies of the winglet geometry impact on its local flow field and how this correlates with overall efficiency. Therefore, the current work is aimed at filling this gap through a quantitative analysis.

In this paper, a brief design space exploration of a wingtip fence and a classical winglet is presented and a detailed study has been performed on the winglet local flow field, where DLR-F4 lift and drag are correlated with changes in the winglet local angle of attack spanwise, which in turn changes according to the aircraft angle of attack and the winglet’s own geometrical parameters, in particular its cant angle. CFD and analytic search for an optimal cant angle has led to considering a spanwise-variable cant (curved) winglet concept, alternative to cant-morphing solutions presented in [

11,

12,

13]. The curved winglet span was parametrized using a second order function, which was in turn optimized for getting the most beneficial local angle of attack distribution. Recently, the concept studied herein, of a curved-span winglet, has become relatively well-established with the introduction of blended winglets by Aviation Partners, Inc. [

9,

22], and Airbus wide body A350 XWB and very lately, the new long range A330 Neo, both featuring a double curvature (spanwise and plan form) winglet [

23,

24]. At the concluding part, a comparison of the static structural stress growth on the wing structure is given for each of the CFD-analyzed wingtip devices, allowing an overall evaluation of their weight penalty, and its juxtaposition to their aerodynamic benefits.

An important contribution of this paper is the front-view parametrization and optimization method used herein for a non-planar lifting surface, applying as an objective function the spanwise local angle of attack distribution. Whereas most parametrization studies for planar wings seek either an optimal plan form, optimal span-wise airfoil shape variation and/or chord-wise airfoil curvature variation (see [

25] for instance), the front projection optimization issue is paramount for non-planar lifting surfaces as wings with winglets, c-wings and closed lifting surfaces, where both the geometric and effective local angles of attack do not change uniformly spanwise, when the aircraft angle of attack changes. Examples of other approaches for winglets parametrization can be found in [

25,

26]. The evolutionary geometry parametrization approach presented by Zingg et al. in [

25] can be applied to parametrize a span-curved winglet geometry by starting from a classical straight winglet then using B-spline approximation to get a generic curved winglet, and by inserting additional knots at the winglet root and/or tip sections, based on gradual inclusion of different design variables. A comprehensive study by Luciano et al. [

27] gives a method for predicting the minimum induced drag of non-planar wings through a configuration-invariant analytic formulation of the unknown circulation distribution. The authors present a numerical tool implementable in MATLAB, which allows the definition of non-planar wings’ optimal circulation distribution with minimum induced drag.

## 4. Discussion of Results and Conclusions

In this study, we presented a solution to a multidisciplinary optimization problem, concerning the interaction of sub- and near transonic aerodynamics with the cantilever of DLR-F4 wing-body prototype, equipped with wingtip devices of different geometries. The solution algorithm begins with low-cost CFD design space exploration on relatively coarse grids, which allows us to evaluate the general aerodynamic efficiency and viability of using different wingtip devices, as well as their most critical geometrical parameters. In the light of CFD results and using simple geometrical analysis, a mathematical model of the local flow field around the winglet and immediately downstream was developed using the local angle of attack function along the winglet span. High fidelity CFD simulations and mathematical analysis are then launched to optimize the chosen critical geometrical parameter (cant angle for instance), while keeping other parameters fixed. Besides lift and drag, CFD simulations yield air pressure distribution over the wing surface, which is used by the stress calculation module as an external load input for numerical modeling of the stress-strain state of the wing cantilever structure. Based on the results of evaluating the growth of lift-to-drag ratio of DLR-F4 after equipping it with different wingtip devices, and its juxtaposing with the increase in structural stress, a tool has been developed for estimating and comparing the overall efficiency through calculating the L/D ratio increment at cruising flight regime, and dividing it by the maximum structural stress increment, encountered during most dangerous off-design flight modes which the wing structure is, nonetheless, supposed to withstand. Using this algorithm, the wingtip device geometry can be optimized in a multidisciplinary fashion at the earliest stages of wing design. As for Whitcomb winglet configurations described in this paper, it has been revealed that the most balanced choice would be the suggested concept of a curved (nearly elliptical) winglet, which provides a compromise solution for cant angle choice.

Future research may enrich the used herein mathematical model of the winglet local flow field by analyzing the impact on its local flow field of other geometrical parameters such as the sweep angle, which may lead to a similar optimal elliptic-leading-edge compromise solution. Combined with a curved span, this curved leading edge would result in a double-curvature winglet, where a three-dimensional parametrization function, depicting a spatial double-curved winglet centerline, is required to be deduced and thoroughly optimized.

The winglet efficiency estimation factor should be extended to include the winglets’ effect on directional stability, by quantifying possible reductions to the vertical tail plan area and its structural weight, as well as by a quantified analysis of winglets’ impact on the wing aeroelasticity and the overall aero-acoustic and environmental footprint of airliners in the long run. In this regard, it is advisable to use, as an experimental platform, a more realistic transport aircraft prototype as DLR-F6 or –F11, that includes what is typical for passenger aircraft high lift devices and engine nacelles.