International Journal of Aerospace Engineering

Volume 2018, Article ID 3104902, 15 pages

https://doi.org/10.1155/2018/3104902

## Prediction of Lift Coefficient for Tandem Wing Configuration or Multiple-Lifting-Surface System Using Prandtl’s Lifting-Line Theory

School of Astronautics, Beihang University, Beijing 100191, China

Correspondence should be addressed to Hao Cheng; nc.ude.aaub@oahgnehc

Received 3 December 2017; Accepted 3 June 2018; Published 8 July 2018

Academic Editor: William W. Liou

Copyright © 2018 Hao Cheng and Hua Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In tandem airfoil configuration or multiple-lifting-surface layouts, due to the flow interaction among their lifting surfaces, the aerodynamic characteristics can be affected by each other. In accordance with Prandtl’s classical lifting-line theory, a method to calculate the section lift coefficient for the tandem wing configuration or multiple-lifting-surface system is presented. In that method, the form of Fourier sine series is used to express the variation of the section circulation which changes continuously along the wingspan. The accuracy of the numerical solutions obtained by the method has been validated by the data obtained from computational fluid dynamics and tunnel experiment. By varying the design parameters, such as the gap, the stagger, the incidence angle, the wingspan, the taper ratio as well as the aspect ratio, a series of tandem wing configurations are tested to analyze the lift coefficient and the induced drag of each lifting surface. From the results, it can be seen that the bigger negative gap and stagger can produce better lift characteristic for tandem wing configuration. Besides, it will also be beneficial for the lift characteristic when the incidence angle and the wingspan of fore wing are appropriately declined or if the incidence angle and the wingspan of hind wing are appropriately increased.

#### 1. Introduction

Compared to the traditional layout, the induced drag for tandem airfoil configuration is smaller and the wingspan can be reduced with the same stiffness, for the reason that the two wings in tandem airfoil configuration are all generating positive lift. Researches on tandem airfoil configuration with its unique aerodynamic advantage were conducted for decades. Bottomley [1, 2] summarized the history of aircrafts with tandem airfoil layouts and pointed out the advantages and disadvantages of that kind of design. To study the aerodynamic characteristics for tandem airfoil configuration, three different methods, including tunnel experiment, computational fluid dynamics (CFD) method, and theoretical calculation, were extensively used.

Through experiments, Feistal et al. [3] conducted experiments on a series of multiwing configuration with finite span wings at Reynolds number of 1.4*e*6, and the results indicated that the interaction between the two wings was favorable to the fore wing (the maximum lift coefficient increased), while the maximum lift coefficient of the hind wing did not decrease. Scharpf and Mueller [4] conducted wind tunnel experiments on different tandem wing configurations, to investigate the influence on the tandem wing aerodynamic characteristics by the horizontal distance, vertical distance, and the decalage between the two wings. Results obtained from tunnel experiments agree well with the real situation but with enormous financial costs.

Another high-fidelity method, CFD, has also been widely used to simulate the aerodynamic characteristics for different layouts. Alley et al. [5] presented a method, which used a correlation obtained from grid-resolved computational fluid dynamics solutions, to predict the maximum lift coefficient for wing of arbitrary planform including the effects of twist and sweep. Fanjoy and Dorney [6] performed a series of numerical experiments, in which a Navier-Stokes analysis adapted for external flow was used to calculate the flow field around the dual-wing geometry and the stagger was varied, to study the tandem-airfoil interaction in different flight regimes. By solving the incompressible Navier-Stokes equation on overlapping grids, Broering and Lian [7] investigated the effect of wing spacing between fore and hind wings on the aerodynamics of a tandem flapping wing configuration numerically at Reynolds number of 5000. Zhang and Yu [8] investigated the unsteady aerodynamics of an aircraft with tandem wing configuration in the morphing stage using CFD methods. Most recently, Patidar et al. [9] investigated the lift and drag characteristics of a Busemann biplane with staggered tandem airfoil configurations at various Mach numbers between 0.5 and 2.5 using computational fluid dynamics. Computational fluid dynamics methods can satisfy the required accuracy with little financial cost but low computational efficiency.

Compared with tunnel experiment and the CFD method, theoretical calculation with little financial cost and high efficiency also has been widely used during the aircraft conceptual design stage. Prandtl’s classical lifting-line theory [10] can be used to predict the section lift coefficient by obtaining an infinite series solution for the spanwise distribution of vorticity generated on the wing, which can compromise between the crude approximations provided by analytic methods as well as low-fidelity analyses and the high level of geometric detail, time, and resources required for the high-fidelity tools, such as Reynolds-averaged Navier-Stokes computational fluid dynamics (CFD) methods, in the early aircraft conceptual design stage. With the assumption of the linear relationship between section lift and section angle of attack as well as the assumption of a straight lifting line, Prandtl’s classical lifting-line theory can provide an analytical solution in the form of an infinite sine series for the circulation distribution to calculate the spanwise distribution of lift acting on a finite-lifting surface. For the evaluation of the coefficients in the sine series, Glauert [11] presented a very straightforward method. Moreover, a more rigorous and more rapidly converging method using Fourier series expansion was developed by Rasmussen and Smith [12]. Meanwhile, McCormick [13] proposed a purely numerical method to solve the lifting-line equation for a single-lifting surface with a straight lifting line, and Anderson and Corda [14] improved the numerical method by relaxing the linear assumption of the relationship between section lift and section angle of attack. However, all the aforementioned methods, to obtain the solution to the classical lifting-line equation, only applied for the particular configuration with a single-lifting surface with no sweep and no dihedral.

For the presence of the flow interaction between the fore wing and the hind wing, the wake generated by the fore wing will be hampered by the hind wing, which will also change the effective angle of attack of the hind wing, so the aerodynamic characteristics of the tandem airfoil configuration are different from the single-wing case. For the tandem wing configuration or the configuration with multiple-lifting surface, Phillips and Snyder [15] presented a numerical lifting-line method, which was based on the Prandtl’s original model of a finite wing, to predict the forces and moments acting on a system of lifting surfaces. In this numerical lifting-line method, the continuous distribution of section circulation was equivalent to a finite number of discrete horseshoe vortices approximately, which was referred to what proposed by Katz and Plotkin [16]. Furthermore, Spall et al. [17] calculated the solutions obtained from a numerical method based on Prandtl’s lifting-line theory, valid for multiple-lifting surface with arbitrary sweep, for a number of rigid wing and geometries. Most recently, Jacobs et al. [18] extended the numerical lifting-line method to transonic speeds and application to multiple-lifting-surface configurations, such as wing-canard configuration and three-lifting-surface business jet. Thus, the accuracy of the methods used to obtain the solution to the lifting-line equation of tandem wing configuration or multiple-lifting-surface configuration depends on the finite number of discrete horseshoe vortices, and the section lift coefficient is not continuous along the wingspan.

In this paper, the main objective is to predict the section lift coefficient for the tandem wing configuration or multiple-lifting-surface system. A lifting-line method based on Prandtl’s classical lifting-line theory is proposed, in which the solution for the spanwise distribution of section circulation is expressed in the form of Fourier series truncated to a finite series. This approach can be used to obtain the lift acting on each lifting surface of tandem wing design and configuration with two lifting surfaces in arbitrary positions. Additionally, a series of tandem wing configuration with different parameters, including the gap, the stagger, the incidence angle, and the wingspan, are calculated and simulated using the proposed method to analyze the effect of the parameters on characteristics including lift and induced drag of each lifting surface. Results with this method are validated by experimental data and CFD results at small angle of attack. The accuracy from the presented method is shown to be comparable to that obtained from experiment and CFD with little experiment cost and computational cost.

#### 2. Lifting-Line Theory for Tandem Wing Configuration

In this section, Prandtl’s classical lifting-line theory for a single finite wing is presented firstly, and then, the lifting-line theory for tandem wing configuration is proposed on the foundation of Prandtl’s classical lifting-line theory combined with the Biot-Savart law.

##### 2.1. Prandtl’s Classical Lifting-Line Theory

Traditionally, the distribution of section circulation along the span for a single finite wing, with no sweep or dihedral and having an arbitrary spanwise variation in chord length, can be obtained by solving Prandtl’s classical lifting-line equation. Prandtl’s lifting-line theory provides a simple and precise spanwise distribution of section circulation for the special case of an elliptic wing. Otherwise, to solve Prandtl’s lifting-line equation for the wing with arbitrary shape, the change of variable is very necessary.

The variation in section circulation along the span of a finite wing, , as approximated by the infinite Fourier sine series solution to Prandtl’s lifting-line equation, can be written as

Based on the Kutta-Joukowski theorem and the definition of the lift coefficient, the following relationship between the section circulation, , and the lift coefficient, , is

Moreover, according to the assumption that the lift can be closely approximated as linear function of angle of attack for airfoil at small angle of attack, the section lift coefficient, , can be approximately calculated as where is the airfoil section lift slope, is the zero-lift angle of attack, and is the effective local angle of attack.

Consequently, combining (3) and (4), the Prandtl’s lifting-line equation applying on each lifting surface can be derived as

Prandtl’s classical lifting-line equation can be solved by calculating the effective local angle of attack, , which can be obtained from the induced velocity generated by its self-downwash effect at a series of control points along the wingspan.

##### 2.2. Application on Tandem Wing Configuration

For the tandem wing configuration, the variation in section circulation along the span of the fore wing and the hind wing, and , as approximated by the infinite Fourier sine series solution to Prandtl’s lifting-line equation, can be, respectively, written as

In order to distinguish the parameters, we defined some subscripts for the variables appeared in the flowing part, which can be explained in Table 1.