1. Introduction
Understanding the aerodynamic properties of a flying vehicle is essential for determining its performance and flight dynamics. These aerodynamic characteristics can be obtained through experimental, computational, and also flight tests. Computational methods can solve the Navier–Stokes equations, including viscosity terms (with high computational cost), or solve the equations with potential models (simplified linearized models). The differential equations used for solving the most relevant problems in low-speed (incompressible) aerodynamics will be a simplified version of the equations governing fluid mechanics, that is, the conservation equations for mass, momentum, and energy (if necessary) [
1].
The majority of fluid studies and their engineering applications require a solution within a fluid domain, usually containing a solid body, with a series of boundaries. Theoretical potential aerodynamics models are a reduction of the general equations, so it is important to understand these introduced simplifications. This allows us to appreciate the power and versatility of the models (both analytical and numerical), as well as the limitations imposed by the simplifications themselves.
It is known [
1] that at high Reynolds numbers
, the dimensions of the boundary layer (inversely proportional to
or
) are small, and the effects of viscosity are confined to this thin region attached to the obstacle. Therefore, potential models will predict lift, induced drag, and aerodynamic moment with good approximation but not viscous drag.
Potential aerodynamics is valid outside the boundary layer [
1,
2], outside the shear layers and wakes, where the viscosity effects are negligible or non-existent; then, outside these regions, the vorticity is zero. The velocity field in potential aerodynamics derives from the potential of velocity
, then
. That is, the assumptions of the flow will be incompressible motion and zero viscosity. Thus, with these approximate models, it will suffice to solve the continuity equation for incompressible flow
and the Euler equation (relationship between pressure
p and velocity
V, i.e., the Bernoulli equation) [
3].
From the theorems of scientist Hermann Von Helmholtz [
3] based on vortices for incompressible flow, we can establish that the intensity of a vortex filament is constant along its length, and it cannot start or end in the fluid. A moving fluid forms a vortex tube, and its intensity remains constant to the extent that the tube moves. The induced velocity by a vortex segment is defined by the Biot–Savart law [
2,
3].
From the aforementioned conditions, it is possible to establish a set of elementary solutions of the Laplace equation
. These elementary solutions can be sources, doublets, and vortices [
1,
2,
3]. Being elementary solutions of the Laplace equation, the principle of superposition can be applied, which forms the foundation for solving the fluid field around complex surfaces. The Rankine oval, flow around a cylinder, and flow over a sphere [
1] are examples of flows around simple surfaces or bodies. With all of this, it is indicated that the solution of potential flow (incompressible, inviscid) around arbitrary bodies can be determined by distributing elementary solutions over the surfaces to be modeled. In other words, the distributions of sources, the distributions of doublets, and the distributions of vortices can be superposed.
The solution of potential flow could be solved using analytical methods, but obtaining the mentioned solutions is only possible in very specific and limited cases. If appropriate simplifications are made on geometric surfaces (curvature, thickness, etc.), it allows linearizing the problem and applying numerical techniques. There are different software that have been developed in the past years for different applications and that solve the aerodynamic problem using the Navier–Stokes equations for potential flow equations [
4,
5]. To simulate using the method of small perturbations on a three-dimensional wing, it could be achieved using distributions of singularities (elementary solutions); a horseshoe vortex distribution forms the basis of the VLM method. This way, problems can be solved on three-dimensional lifting surfaces with dihedral, sweep, and even sideslip. This is a generalized solution of Prandtl’s lifting-line theory.
TORNADO uses the vortex lattice method (VLM), which is an aerodynamic analysis at steady state. AVL [
6], Surfaces [
7], XFLR5 [
4] and VSPAERO [
8] are also vortex lattice-based and can give good results for preliminary studies. Some articles show the significant evolution and application capacity of these methods, for example, using non-linear VLM [
9,
10], unsteady VLM (UVLM) [
5] or even adapting VLM to supersonic aircraft [
11]. From this point on, the focus will be on TORNADO, which was the tool chosen to develop the code due to the simplicity of adapting the program. The wing is supposed to be divided into trapezoidal panels (see
Figure 1), each of them with an associated horseshoe vortex. The turbillonary field generated by the vortices is divided in two regions: the free vortices, which cause the induced velocity, and the bound vortices, which are related to the airfoil lift effect. The panels have two important points, the control point in 3/4 of the wing chord (c) and the head of the vortex in 1/4c. Each vortex has a circulation distribution (associated with the induced velocity and the interaction between vortices), which allows to calculate the lift distribution. With that distribution and the angle of attack, the lift and induced drag coefficient can be obtained. The VLM method is one of the most extended methods to calculate aerodynamics properties based on potential features of the flow, that is, lift force, moment and induced drag. The computational effort is extraordinarily low due to the linearized nature of the equations, as well as the final result of solving linear systems of equations (even of large dimensions). According to this, the VLM method has limitations, as it can be applied when low angles of attack are considered (to avoid effects of stall conditions) to ignore the compressibility effects, and it is a method by which any type of drag dependent on viscosity cannot be calculated [
12].
The wings to be studied in this article are those that introduce wingtip devices. Winglets are becoming increasingly important elements in aircraft design, as operating costs and environmental concerns are on the rise. Therefore, from an aerodynamic point of view, wingtip devices are able to reduce induced drag [
13] and increase aerodynamic efficiency.
Figure 2 presents different configurations that have been used or are in development.
Fredick W. Lanchester was the first person to mention the possibility of creating a vertical surface (end-plate), in the belief that it would reduce drag by controlling wing tip vortices [
14]. Later, it was Richard T. Whitcomb who developed the first winglet, with the inspiration of bird wings [
15]. One of the conclusions was that the winglets’ presence helped at a Mach number of 0.78, and keeping the lift coefficient from the state without winglets decreased the induced drag about 20% and increased the aerodynamic efficiency about 9% [
15]. Another important contribution was made by John M. Kuhlman, who realized a study about the effects that winglets have in low elongation configurations. He concluded that the reduction in the induced drag depends solely on the ratio of the winglet length to the wingspan [
16].
Figure 3 shows some winglet examples that are designed with the developed algorithm, such as the blended winglet [
17], the spiroid winglet [
18,
19,
20,
21,
22,
23] and a first approximation of the wing grid [
24,
25,
26,
27]. They were selected due to the previous investigations conducted in the past years. It is worth noting that companies have already implemented the blended winglet and the grid wing. The blended winglet is the main design adopted by the aeronautical industry for commercial airplanes. Top companies, such as Airbus and Boeing, have implemented those designs for new aircraft models, with satisfactory results in terms of efficiency and fuel consumption [
28].
This paper presents an algorithm that facilitates the wing geometry design to complement the use of TORNADO. The aim of this article is to realize a first iteration of the geometry design of a winglet for its implementation in TORNADO. This software is an extremely valuable tool that offers first-order approximations of performance and lower computational cost than CFD, expediting the optimization of geometry design.
4. Conclusions
Although TORNADO encounters challenges in two specific cases (the study of the separation point and the overall calculation of the drag coefficient), it is a very useful tool, as evidenced in this article, and the winglet primarily affects induced drag (the studied one). The algorithm implementation achieved the aim of the article: to make an algorithm that can create complex geometry in TORNADO using a basic data entry mechanism and to obtain the optimal geometry with very low computational cost before using CFD. Moreover, the algorithm allows the automation of the winglet generation process selecting basic wing characteristics, such as the root and tip chord, the airfoil, tape, dihedral, the winglet type, panel distribution, number of joints, and the according radius, among others.
In addition, in order to complete the drag reduction study, TORNADO has the capability to calculate the lift and induced drag coefficients. This may lead to the development of a new algorithm (e.g., genetic algorithm) to determine the optimal winglet during the initial design. The final winglet design could then be subject to CFD and experimental analysis for validation. Therefore, this article provides a solid basis for the subsequent phases of winglet development.