Nomenclature

${A_j}$

Fourier coefficients used in the solution to the fundamental lifting-line equation

${a_j}$

decomposed Fourier coefficients corresponding to wing planform shape

${B_j}$

normalised Fourier coefficients: ${B_j} \equiv {\rm{\;}}{A_j}/{A_1}$

$b$

wingspan

${b_j}$

decomposed Fourier coefficients corresponding to spanwise symmetric twist

${\tilde C_{L,\alpha }}$

section lift slope

${C_{{D_i}}}$

induced-drag coefficient

${C_L}$

lift coefficient

${C_\ell }$

rolling-moment coefficient

${C_n}$

yawing-moment coefficient

$c$

local section chord length

${c_j}$

decomposed Fourier coefficients corresponding to spanwise antisymmetric twist caused by ailerons

$j$

term in the Fourier sine series

$N$

number of terms retained when truncating the infinite Fourier sine series

${R_A}$

aspect ratio

${R_{n/\ell }}$

roll-yaw control ratio

${R_T}$

taper ratio

$s$

normalised spanwise coordinate: $s = 2z/b$

${s_{{\rm{centre}}}}$

spanwise position of the aileron centre

${s_{{\rm{root}}}}$

aileron root, spanwise position of the inboard aileron edge

${s_{{\rm{tip}}}}$

aileron tip, spanwise position of the outboard aileron edge

${s_{{\rm{width}}}}$

aileron width along the semispan

${V_\infty }$

freestream velocity

$z$

spanwise coordinate centred at the wing root, positive out the left wing

Greek symbol

$\alpha $

local geometric angle-of-attack relative to the freestream

${\alpha _{L0}}$

local zero-lift angle-of-attack

${\rm{\Gamma }}$

local circulation strength on the lifting line

${\delta _a}$

amount of aileron deflection

${\varepsilon _f}$

local section flap effectiveness

$\theta $

change of variables for the spanwise coordinate

$\chi $

normalised spanwise antisymmetric twist distribution function used to describe aileron size and location

${\rm{\Omega }}$

amount of the symmetric component of wing twist

$\omega $

normalised spanwise symmetric twist distribution function

2.0 Analytic formulation

Prandtl’s classical lifting-line theory [Reference Prandtl17, Reference Prandtl18] has formed the basis of our understanding of wing design for the past century. Solutions to the fundamental lifting-line equation are obtained by assuming a circulation distribution ${\rm{\Gamma }}$ that can be written in terms of a Fourier sine series as

(1) \begin{align}{\rm{\Gamma }}\!\left( \theta \right) = 2b{V_\infty }{A_1}\!\left( {{\rm{sin}}\!\left( \theta \right) + \mathop {\mathop \sum \limits_{j = 2} }\limits^\infty {B_j}\ \rm{sin}\ j\theta } \right),\theta \equiv {\rm{co}}{{\rm{s}}^{ - 1}}\!\left( { - 2z/b} \right),{B_j} \equiv {A_j}/{A_1}\end{align}

where $b$ is the wing span, ${V_\infty }$ is the freestream velocity, ${A_j}$ are the Fourier coefficients, ${B_j}$ are normalised Fourier coefficients and $\theta $ is a change of variables related to the spanwise location $z$ . The Fourier coefficients ${A_j}$ are obtained by forcing the lifting-line equation to be satisfied at a finite number of $\left( N \right)$ locations along the wing. This gives a linear system of equations that can be solved for the truncated Fourier series

(2) \begin{align}\mathop {\mathop \sum \limits_{j = 1} }\limits^N {A_j}\!\left( {{{4b} \over {{{\tilde C}_{L,\alpha }}c\!\left( \theta \right)}} + {j \over {{\rm{sin}}\!\left( \theta \right)}}} \right){\rm{sin}}\!\left( {j\theta } \right) = \alpha \!\left( \theta \right) - {\alpha _{L0}}\!\left( \theta \right)\end{align}

where ${\tilde C_{L,\alpha }}$ is the aerofoil section lift slope, $c$ is the local section chord, $\alpha $ is the local section geometric angle-of-attack, and ${\alpha _{L0}}$ is the local section zero-lift angle-of-attack. The integrated lift, induced-drag, rolling-moment and yawing-moment coefficients on the wing in the absence of rolling rate can be computed from the resulting Fourier coefficients [Reference Phillips26, Reference Phillips27]

(3) \begin{align}{C_L} = \pi {R_A}{A_1}\end{align}

(4) \begin{align}{C_{{D_i}}} = {{C_L^2} \over {\pi {R_A}}}\!\left( {1 + \mathop {\mathop \sum \limits_{j = 2} }\limits^N jB_j^2} \right)\end{align}

(5) \begin{align}{C_\ell } = - {1 \over 4}{C_L}{B_2} = - {1 \over 4}\pi {R_A}{A_2}\end{align}

(6) \begin{align}{C_n} = - {{C_L^2} \over {4\pi {R_A}}}\!\left( {\mathop {\mathop \sum \limits_{j = 2} }\limits^N \!\left( {2j - 1} \right){B_{j - 1}}{B_j}} \right)\end{align}

where ${R_A}$ is the wing aspect ratio defined as ${R_A} \equiv {b^2}/{S_w}$ and ${S_w}$ is the wing area.

An alternative form of lifting-line theory has been presented [Reference Phillips26] that decouples the effects of angle-of-attack, washout and aileron deflection. This method has been used to develop significant insight into the optimum washout distributions and will be used here to consider optimum aileron placement. In a theoretical sense, the deflection of an aileron is simply a step-change in aerodynamic twist, since a deflection changes the section zero-lift angle-of-attack but produces very little change in the section lift slope. Here we apply the change of variables

(7) \begin{align}{B_j} = {{\pi {R_A}} \over {{C_L}}}\left[{a_j}\!\left( \alpha - {\alpha _{L0}}\right)_{{\rm{root}}} + {b_j}{\rm{\Omega }} + {c_j}{\delta _a} \right]\end{align}

where ${a_j}$ , ${b_j}$ , and ${c_j}$ are decomposed Fourier coefficients, ${\rm{\Omega }}$ is the amount of symmetric wing twist, and ${\delta _a}$ is the aileron deflection, or the amount of antisymmetric twist. The three components on the right-hand side of Equation (7) are the contributions of angle-of-attack and planform, spanwise-symmetric twist (washout) and spanwise-antisymmetric twist caused by an aileron deflection. Considering only wings with symmetric planforms and noting from Equation (2) that only odd Fourier coefficients depend on symmetric components and only even coefficients depend on antisymmetric components, Equation (7) can be expressed as

(8) \begin{align}{B_j} = {{\pi {R_A}} \over {{C_L}}}\!\left\{ \begin{array}{cc} {{a_j}{{(\alpha - {\alpha _{L0}})}_{{\rm{root}}}} + {b_j}{\rm{\Omega }}} & {{\rm{\;\;\;\;\;}}j{\rm{\;odd}}} \\[5pt] {{c_j}{\delta _a}} & {{\rm{\;\;\;\;\;}}j{\rm{\;even}}}\end{array}\right.\end{align}

The decomposed Fourier coefficients ${a_j}$ , ${b_j}$ and ${c_j}$ can be obtained from

(9) \begin{align}\mathop {\mathop \sum \limits_{j = 1} }\limits^N {a_j}\!\left( {{{4b} \over {{{\tilde C}_{L,\alpha }}c\!\left( \theta \right)}} + {j \over {{\rm{sin}}\!\left( \theta \right)}}} \right){\rm{sin}}\!\left( {j\theta } \right) = 1\end{align}

(10) \begin{align}\mathop {\mathop \sum \limits_{j = 1} }\limits^N {b_j}\!\left( {{{4b} \over {{{\tilde C}_{L,\alpha }}c\!\left( \theta \right)}} + {j \over {{\rm{sin}}\!\left( \theta \right)}}} \right){\rm{sin}}\!\left( {j\theta } \right) = \omega \!\left( \theta \right)\end{align}

(11) \begin{align}\mathop {\mathop \sum \limits_{j = 1} }\limits^N {c_j}\!\left( {{{4b} \over {{{\tilde C}_{L,\alpha }}c\!\left( \theta \right)}} + {j \over {{\rm{sin}}\!\left( \theta \right)}}} \right){\rm{sin}}\!\left( {j\theta } \right) = {\varepsilon _f}\!\left( \theta \right)\chi \!\left( \theta \right)\end{align}

where $\omega \!\left( \theta \right)$ is a normalised symmetric twist distribution function (washout distribution), $\chi \!\left( \theta \right)$ is a normalised antisymmetric twist distribution function caused by the ailerons, and ${\varepsilon _f}\!\left( \theta \right)$ is the section flap effectiveness of the ailerons [Reference Phillips27].

We will limit this study to a special class of optimum symmetric lift distributions given by ${B_j} = 0$ for all odd terms $j \gt 3$ [Reference Hunsaker, Montgomery and Joo7, Reference Hunsaker, Moorthamers and Joo9, Reference Phillips, Hunsaker and Joo11–Reference Taylor and Hunsaker14, Reference Prandtl16]. This class of lift distributions provides the minimum induced drag for various aerodynamic and structural design constraints [Reference Phillips, Hunsaker and Joo11–Reference Taylor and Hunsaker14, Reference Prandtl16–Reference Prandtl18] and is fully defined by the single parameter ${B_3}$ . For example, an elliptic lift distribution is defined by ${B_3} = 0$ , and Prandtl’s bell-shaped lift distribution is defined by ${B_3} = - 1/3$ . Other optimal lift distributions minimising induced drag for various design and operating constraints exist within the range $ - 1/3 \lt {B_3} \lt 0$ and are shown in Table 1 along with their associated structural and aerodynamic constraints [Reference Phillips, Hunsaker and Joo11–Reference Taylor and Hunsaker14, Reference Prandtl16–Reference Prandtl18, Reference Phillips26].

Table 1. Optimal lift distributions for various structural constraints

For a wing of arbitrary planform to produce a prescribed lift coefficient and lift distribution within this class of optimal lift distributions, the symmetric dimensionless twist distribution, twist amount and angle-of-attack must be [Reference Phillips and Hunsaker12]

(12) \begin{align}\omega \!\left( \theta \right) = {{4b/{{\tilde C}_{L,\alpha }}\left[ \left( {1 - {B_3}} \right)/{c_{{\rm{root}}}} - \right[{\rm{sin}}\!\left( \theta \right) + {B_3}\ {\rm{sin}}\!\left( {3\theta } \right)\left] {/c\!\left( \theta \right)} \right] - 3{B_3}\!\left[ {1 + {\rm{sin}}\!\left( {3\theta } \right)/{\rm{sin}}\!\left( \theta \right)} \right]} \over {4b\!\left( {1 - {B_3}} \right)/\left( {{{\tilde C}_{L,\alpha }}{c_{{\rm{root}}}}} \right) - 12{B_3}}}\end{align}

(13) \begin{align}{\rm{\Omega }} = {{{C_L}} \over {\pi {R_A}}}\!\left( {{{4b\!\left( {1 - {B_3}} \right)} \over {{{\tilde C}_{L,\alpha }}{c_{{\rm{root}}}}}} - 12{B_3}} \right)\end{align}

(14) \begin{align}{(\alpha - {\alpha _{L0}})_{{\rm{root}}}} = \frac{{{C_L}}}{{\pi {R_A}}}\!\left( {\frac{{4b\!\left( {1 - {B_3}} \right)}}{{{{\tilde C}_{L,\alpha }}{c_{{\rm{root}}}}}} + 1 - 3{B_3}} \right)\end{align}

The normalised twist distribution function due to aileron placement can be written as [Reference Phillips27]

(15) \begin{align}\chi \!\left( s \right) \equiv \left( {\begin{array}{r@{\quad}l}0, & {}{s \lt - {s_{{\rm{tip}}}}}\\1, & {}{ - {s_{{\rm{tip}}}} \le s \lt - {s_{{\rm{root}}}}}\\0, & {}{ - {s_{{\rm{root}}}} \le s \le {s_{{\rm{root}}}}}\\ - 1, & {}{{s_{{\rm{root}}}} \lt s \le {s_{{\rm{tip}}}}}\\0, & {}{{s_{{\rm{tip}}}} \lt s}\end{array}} \right.\end{align}

where $s$ is a normalised spanwise coordinate $s \equiv 2z/b$ , and ${s_{{\rm{root}}}}$ and ${s_{{\rm{tip}}}}$ are the spanwise locations corresponding to the aileron edges closest to the wing root and wing tip and will be called the aileron root and aileron tip respectively. These two design variables ${s_{{\rm{root}}}}$ and ${s_{{\rm{tip}}}}$ can be altered to consider their effect on roll-yaw coupling for this class of symmetric lift distributions.

From Equations (5) and (8) it can be seen that only the ${c_2}$ component of the aileron contributes to the rolling-moment coefficient

(16) \begin{align} C_\ell = - \frac{1}{4}\pi {R_A}{c_2}{\delta _a}\end{align}

By inspection of Equations (4), (8), and (11), the $\chi\!\left( \theta \right)$ design that minimises induced drag requires ${c_j} = 0$ for $j \gt 2$ [Reference Hunsaker, Montgomery and Joo10]. Such a design requires continuous antisymmetric twist [Reference Joo, Marks, Zientarski and Culler28–Reference Molinari, Arrieta and Ermanni39] and is not possible with the discrete aileron deflection distribution defined in Equation (15). Effects of discrete aileron placement on induced drag have been considered by Feifel [Reference Feifel40] and Brincklow et al. [Reference Brincklow and Hunsaker41]. From Equation (6) we see that the yawing-moment depends on the product of neighbouring Fourier coefficients ${B_{j - 1}}{B_j}$ . This fact combined with the definition of the class of symmetric lift distributions considered in this work means that only the ${c_2}$ and ${c_4}$ components of the aileron contribute to the yawing-moment coefficient. From Equations (6) and (8) the yawing-moment coefficient can be written as

(17) \begin{align}{C_n} = \frac{1}{4}{C_L}{\delta _a}\!\left[ {\left( {3 + 5{B_3}} \right){c_2} + 7{B_3}{c_4}} \right]\end{align}

Equations (16) and (17) can be combined to give the roll-yaw control ratio [Reference Hunsaker, Montgomery and Joo7]

(18) \begin{align}{R_{n/\ell }} \equiv \frac{{{C_n}}}{{{C_L}{C_\ell }}} = - \frac{1}{{\pi {R_A}}}\left( {3 + 5{B_3} + 7{B_3}\frac{{{c_4}}}{{{c_2}}}} \right)\end{align}

Equation (18) provides important insight into the design of wings for prescribed roll-yaw coupling. Notice that for ${B_3} = 0$ (the elliptic lift distribution), the roll-yaw coupling will always be negative and equal to

(19) \begin{align}{R_{n/\ell }} = \frac{{ - 3}}{{\pi {R_A}}}\end{align}

This relationship was first published by Munk [Reference Munk42] and demonstrates that the elliptic lift distribution will always produce adverse yaw, independent of the aileron design or placement. However, Equation (18) also shows that for any lift distribution defined by a non-zero value for ${B_3}$ , the roll-yaw coupling depends on the ratio of the decomposed Fourier coefficients ${c_4}/{c_2}$ , which are both related to the aileron design. For ${B_3} \lt 0$ , if ${c_4}/{c_2} \lt - \left( {3 + 5{B_3}} \right)/\left( {7{B_3}} \right)$ , the wing will produce adverse yaw; if ${c_4}/{c_2} = - \left( {3 + 5{B_3}} \right)/\left( {7{B_3}} \right)$ , the wing will produce neutral yaw, and if ${c_4}/{c_2} \gt - \left( {3 + 5{B_3}} \right)/\left( {7{B_3}} \right)$ , the wing will produce proverse yaw. The decomposed Fourier coefficients ${c_2}$ and ${c_4}$ are not independent parameters that can be varied directly. Rather they depend on the wing planform and aileron design through Equations (11) and (15). Even so, Equation (18) provides a strong theoretical foundation demonstrating that proverse yaw is possible for many lift distributions with proper aileron design.

Equation (18) provides additional important insight into the dependence of the roll-yaw control ratio on wing design and operating conditions. Note that the right-hand side of Equation (18) is completely independent of the wing operating condition, including angle-of-attack, lift coefficient or rolling-moment coefficient. Additionally, from Equation (11), it can be shown that ${c_2}$ and ${c_4}$ are both dependent on aileron design, but independent of aileron deflection magnitude. Hence the roll-yaw control ratio is independent of the aileron deflection magnitude. The right-hand side of Equation (18) does depend on the wing aspect ratio, lift distribution ${B_3}$ , wing planform and aileron design. In the following analysis, operating conditions and aileron deflections were chosen out of necessity to produce solutions. As will be illustrated, the results for the roll-yaw control ratio shown for a range of wing planforms, aspect ratios, lift distributions and aileron designs in the following sections are independent of operating condition or aileron deflection magnitude below stall.

Numerical methods could be employed to solve the infinite-series solution to the lifting-line development shown here and correlate the wing and aileron geometry to solutions for ${c_2}$ and ${c_4}$ . However, this classical lifting-line development is best solved using cosine clustering, which is limited in how well the discrete changes in twist due to aileron deflection can be resolved [Reference Brincklow and Hunsaker41]. Therefore, in the remainder of this work, a modern numerical lifting-line algorithm developed by Phillips and Snyder [Reference Phillips and Snyder43] is used, which is a numerical analogue to the classical lifting-line theory but allows for grid clustering in areas of high gradients. In this way, the step-change in twist distribution due to aileron deflection can be properly resolved.

3.0 Numerical method

The numerical lifting-line algorithm of Phillips and Snyder [Reference Phillips and Snyder43], similar to classical lifting-line theory, assumes that the section lift coefficient at any given spanwise location along the lifting surface is equivalent to the 2D case at the same local angle-of-attack. Because of this, the algorithm works best for wings with aspect ratio of four or greater, where spanwise flow effects are minimal. The numerical algorithm shows good agreement with higher fidelity aerodynamic tools and experimental results [Reference Phillips and Snyder43], and has been implemented in several journal publications [Reference Hunsaker, Montgomery and Joo10, Reference Brincklow and Hunsaker41, Reference Phillips, Hansen and Nelson44–Reference Hodson, Hunsaker and Spall49]. The particular implementation of the numerical lifting-line algorithm used in this work is an open-source tool called MachUp [Reference Phillips and Snyder43, Reference Hodson, Hunsaker and Spall49], which allows control points to be clustered at the wing tips as well as near the aileron edges. MachUp was used here to evaluate the effect of aileron placement on the resulting yawing moment for a large range of planforms and ${B_3}$ lift distributions.

A grid resolution study was performed for the case of ${R_A} = 8$ , ${R_T} = 1$ , ${B_3} = - 1/3$ , ${C_L} = 0.5$ , ${C_\ell } = 0.1$ , ${s_{{\rm{root}}}} = 0.5$ , and ${s_{{\rm{tip}}}} = 0.9$ . Figure 1 shows the induced drag coefficient and roll-yaw control ratio results for this study for increasing node count. For reference, the classical lifting-line results are also shown for varying truncation points of the infinite series solution. Results from this study show that using 100 nodes along each wing semispan in the MachUp simulations gives grid-resolved solutions. For this study and the following results, the nodes were distributed using cosine clustering across the inboard section of the wing, the aileron section, and the outboard section. The number of nodes used along each wing section was proportional to the spanwise length of that section divided by the wing semispan. For example, if the aileron covered 20% of the spanwise length of the wing, 20 nodes were used on that section. The aileron flap-chord fraction is unity for all cases in this study. However, as Feifel [Reference Feifel40] notes, the flap-chord fraction does not affect the induced drag predicted by potential-flow algorithms, and only affects the aileron deflection magnitude. As discussed earlier and demonstrated in Equation (18), the aileron deflection has no influence on the roll-yaw control ratio.

Figure 1. Grid resolution of MachUp and classical lifting-line theory.

Figure 2. Induced-drag and roll-yaw control ratio contours.

The class of lift distributions included in this study are completely defined by the single parameter ${B_3}$ . For a given planform, lift coefficient, rolling-moment coefficient and lift distribution defined by ${B_3}$ , the washout distribution, washout magnitude, and root angle-of-attack were computed from Equations (12)–(14) and employed in the MachUp simulation. A Newton-Secant method was used to find the aileron deflection that would produce the prescribed rolling moment, and the resulting induced drag and yawing moment as predicted by MachUp were recorded. From these results, the roll-yaw control ratio ${R_{n/\ell }}$ was computed from the definition given in (18). This process was repeated over a wide range of wing planform shapes and aileron placement. For the results presented in this study, ${\varepsilon _f}\left( \theta \right)$ was set to 1.0 for all wing sections with an aileron, and the section lift slope was set to the theoretical value from thin-aerofoil theory, ${\tilde C_{L,\alpha }} = 2\pi $ .

Figure 2 shows example results for the induced-drag coefficient and roll-yaw control ratio for a rectangular wing with ${R_A}$ = 8, ${B_3}$ = –1/3, ${C_L}$ = 0.5, and ${C_\ell }$ = 0.1. Figure 2 includes two plots that show contours of the induced-drag coefficient and roll-yaw control ratio as a function of the aileron spanwise size and location. Figure 2(a) shows the view with respect to ${s_{{\rm{root}}}}$ and ${s_{{\rm{tip}}}}$ while Fig. 2(b) shows the data with respect to the aileron centre ${s_{{\rm{centre}}}} = \left( {{s_{{\rm{root}}}} + {s_{{\rm{tip}}}}} \right)/2$ and aileron width ${s_{{\rm{width}}}} = {s_{{\rm{tip}}}} - {s_{{\rm{root}}}}$ . Note that the shaded regions are blank since they represent infeasible aileron designs. All results only assume inviscid conditions and do not include the effects of stall. As will be shown, these results are independent of aileron deflection and operating condition below stall. Notice the nearly vertical contour with a noted ${R_{n/\ell }} = 0$ near a ${s_{{\rm{centre}}}}$ of 0.825 in Fig. 2(b). This contour line represents the range of aileron widths that can be used to produce zero roll-yaw coupling, and indicates that an aileron centred at the spanwise location $s = 0.825$ will produce neutral yaw for this design. In general, an aileron with the centre inboard of this location will produce adverse yaw, and an aileron with the centre outboard of this location will produce proverse yaw, regardless of aileron width. The contours for induced drag demonstrate that in general, the induced drag decreases with increasing aileron size along lines of constant roll-yaw control ratio. Therefore, if we wish to minimise induced drag, we should select an aileron that extends to the wing tip. The case that corresponds to this minimum-induced-drag solution for a design with ${R_{n/\ell }} = 0$ is marked with a circle in Fig. 2.

Figure 3 shows similar data based on the planform of the Prandtl-D aircraft created by NASA [Reference Bowers, Murillo, Jensen, Eslinger and Gelzer6] with ${R_A} = 15.55$ and a taper ratio ${R_T} = 0.26$ . NASA’s Prandtl-D wing is an experimental aircraft, which showed Prandtl’s bell-shape wing lift distribution combined with careful aileron design is capable of producing proverse yaw. Research [Reference Bowers, Murillo, Jensen, Eslinger and Gelzer6] supports that this lift distribution is more similar to avian flight compared to the elliptical lift distribution. The design of the aircraft planform is a quarter-scale of the Horten H Xc aircraft; the aircraft has no vertical surfaces, and has elevons starting at 86% of the wing semi-span and continuing to the wing tip [Reference Bowers, Murillo, Jensen, Eslinger and Gelzer6]. For more information about the design, manufacturing, flight-testing and results of this aircraft, the reader is referred to the following publications [Reference Bowers, Murillo, Jensen, Eslinger and Gelzer6, 50, Reference Callen51]. This can be examined in a theoretical sense by considering the data in Fig. 3. The point marked with an ‘x’ represents the aileron design used on the Prandtl-D aircraft. Results shown in this figure predict that the aileron placement on the Prandtl-D will in fact produce proverse yaw and minimise induced drag for that roll-yaw control ratio. Additionally, the aileron design that would produce zero roll-yaw coupling and minimise induced drag is marked with a circle.

Figure 3. Induced-drag and roll-yaw control ratio contours for a Prandtl-D-like wing planform [Reference Bowers, Murillo, Jensen, Eslinger and Gelzer6].

Equation (18) shows that the roll-yaw control ratio is theoretically independent of operating condition, including lift coefficient, rolling moment and aileron deflection magnitude. To verify this numerically, a single wing planform and lift distribution were chosen, and studied over a range of operating conditions. The wing planform and lift distribution used in this study were the same as those used to produce the results in Fig. 2. Two aileron designs were considered for this wing planform and lift distribution. The first was a somewhat arbitrary aileron design defined by ${s_{{\rm{root}}}} = 0.5$ and ${s_{{\rm{tip}}}} = 0.9$ . From the results in Fig. 2, that aileron design was expected to produce adverse yaw. The second aileron design was specifically chosen as the design that produces a neutral roll-yaw control ratio and minimises induced drag, and is defined by ${s_{{\rm{root}}}} = 0.663$ and ${s_{{\rm{tip}}}} = 1.0$ . For each aileron design, the operating lift coefficient ${C_L}$ was varied from 0.001 to 1.0 using 10 equally spaced points, and the operating rolling-moment coefficient ${C_\ell }$ was varied from 0.001 to 0.1 using 10 equally spaced points. Each combination of lift coefficient and rolling-moment coefficient was considered as an individual case, which gave 100 total cases for each aileron design. In each case, the required aileron deflection was found and the resulting yawing moment and corresponding roll-yaw control ratio was obtained. Figure 4 shows results of the study. Notice that the roll-yaw control ratio for each aileron design is independent of the operating condition and aileron deflection. This substantiates the theoretical solution given in Equation (18). Also included in the figure for comparison are the analytic results for an elliptic lift distribution on a wing with the same aspect ratio, as given in Equation (19).

Figure 4. Roll-yaw control ratio results for a range of operating conditions.

4.0 Results

Using the numerical algorithm presented in Section 3 and our understanding of the aileron design space, we now consider the aileron design required to produce a desired roll-yaw coupling over a range of planforms and symmetric lift distributions. To be able to predict the roll-yaw coupling of a given aileron geometry, we simply need to find the aileron design that will produce zero roll-yaw coupling. Any aileron inboard of that location will produce adverse yaw, and any aileron outboard of that location will produce proverse yaw. Since, in general, there are multiple solutions for aileron designs that produce zero roll-yaw coupling for a given wing planform and lift distribution, we wish to select the aileron design that will minimise induced drag. Notice that for the cases shown in Figs. 2 and 3, the induced drag is minimised by selecting an aileron that extends to the tip, i.e. ${s_{{\rm{tip}}}} = 1$ . Repeating this analysis for wing designs in the range: $ - 0.5 \le {B_3} \lt 0,4 \le {R_A} \le 20,0 \le {R_T} \le 1$ and various ${C_\ell }$ values, it was determined that each aileron design within this range that gives zero roll-yaw control ratio and minimum induced drag has ${s_{{\rm{tip}}}} = 1$ . Therefore, in the results that follow, the aileron tip was fixed at the tip, and only the aileron root, ${s_{{\rm{root}}}}$ , was allowed to vary.

The aileron root position that would provide a zero roll-yaw control ratio for each case was found using the gradient-based optimisation code Optix [Reference Hodson, Hunsaker and Spall49], which employs the BFGS algorithm [Reference Broyden52–Reference Shanno55]. Because MachUp uses a finite number of control points per semispan, with small changes to ${s_{{\rm{root}}}}$ , the distribution of control points between the ailerons and regular wing sections can change causing step changes in the solution. This was resolved with an iterative process of running Optix with a fixed nodal distribution between aileron sections and regular wing sections. Once a converged solution was obtained, the nodal distribution was altered to reflect the new ratio of aileron span to wing span, and the optimisation code was run again. This process was repeated in an outer loop until the location of the aileron stopped changing. The optimisation solutions that resulted in a final roll-yaw control ratio with magnitude greater than $1.0E - 9$ were considered as designs that have no solution, and were discarded. In those cases, for the given ${B_3}$ and planform there is not an aileron design with ${\varepsilon _f}\!\left( \theta \right) = 1$ of the form in Equation (15) that can give a neutral roll-yaw control ratio. These scenarios were only encountered for ${B_3}$ values near 0. It is possible that other aileron designs, like multi-segmented ailerons, which have more degrees of freedom, could achieve zero roll-yaw control ratio at ${B_3}$ values nearer to 0. This is because an aileron design with more degrees of freedom would be better able to control the term ${c_4}/{c_2}$ in Equation (18). As the degrees of freedom of the aileron increase, $\varepsilon \!\left( \theta \right)\chi \!\left( \theta \right)$ would approach the absolute optimal solution for minimum induced drag for a given roll-yaw control ratio [Reference Hunsaker, Montgomery and Joo7].

Resulting solutions for the aileron centre locations that give a zero roll-yaw control ratio are shown in Fig. 5 as a function of ${B_3}$ over a wide range of aspect ratios and taper ratios at the condition ${C_L} = 0.5$ and ${C_\ell } = 0.01$ . Neutral lines for wings of aspect ratios from 4 to 20 in increments of 2 are shown, where the neutral line represents a band of designs providing neutral roll-yaw coupling ${R_{n/\ell }} = 0$ with ${s_{{\rm{tip}}}} = 1$ .

Figure 5. Aileron centre location shown as a function of ${B_3}$ for neutral roll-yaw coupling at ${s_{{\rm{tip}}}} = 1$ .

Any aileron design centre below the neutral line shown in Fig. 5 will produce adverse yaw, while any aileron design centre above the neutral line will produce proverse yaw. In all cases, it is assumed that the aileron extends to the wing tip. Note that as the value of ${B_3}$ decreases, solutions that produce proverse yaw can be found further inboard. Also note that as the ${B_3}$ value approaches zero (the elliptic lift distribution), solutions no longer exist. The ${B_3}$ values marked by the vertical lines labeled (a)–(d) correspond to the values in Table 1.

From Fig. 2(a) and (b) we observed that the contours of neutral roll-yaw coupling were nearly vertical when plotted as a function of the aileron centre. Therefore, if the designer wishes to design an aileron to control roll-yaw coupling and does not wish to extend the aileron to the wing tip, this can be achieved as long as the centre of the aileron remains in the correct location. The reader is reminded that maximising aileron width while not changing the aileron centre location will minimise induced drag caused by the ailerons.

5.0 Conclusion

The present work shows that it is possible to design for a certain roll-yaw control ratio through specific spanwise symmetric twist and aileron design without the need of aileron-rudder mixing, differential aileron deflection or Frise ailerons. With proper design, a wing could be developed to give neutral or proverse yaw when the ailerons are deflected. The present study has been limited to wings employing a special class of symmetric lift distributions fully defined by the parameter ${B_3}$ . This class of lift distributions includes multiple analytic solutions that minimise induced drag from the wing when structural and design constraints are considered. For an arbitrary wing planform and desired lift distribution defined by ${B_3}$ , the twist distribution and angle-of-attack that achieves this desired lift distribution is given by Equations (12)–(14). The resulting roll-yaw control ratio for a given planform and lift distribution depends on the aileron placement. However, it was found that any wing employing the elliptic lift distribution will produce adverse yaw independent of the aileron design. Example design spaces for aileron parameters and the resulting roll-yaw control ratio and induced drag are shown in Figs 2 and 3. It was found that the aileron designs that minimise induced drag for a given roll-yaw control ratio generally extend to the wing tip.

An optimisation routine was used in conjunction with a numerical lifting-line algorithm to find the aileron design that would minimise induced drag and ensure a roll-yaw control ratio of zero for a wide range of planforms and lift distributions within this class of distributions. The aileron centre was recorded for each solution, and plotted in Fig. 5, which shows aileron designs as a function of ${B_3}$ for various wing planforms that give zero roll-yaw control ratio while minimising induced drag. For the design space shown in Fig. 5, the aileron designs that minimised induced drag all extend to the wing tip. Any aileron that extends to the tip and has a centre location further outboard than the neutral line shown in Fig. 5 will produce proverse yaw, while a design with a centre location further inboard than the neutral line will produce adverse yaw.

This method of aileron and wing design seem particularly well suited to sailplanes and gliders, which require extreme aerodynamic efficiency while overcoming challenges caused by high-aspect-ratio wings. Some of these challenges are balancing structure weight with allowable wing bending and overcoming significant adverse yaw. The special class of lift distributions included in this study includes solutions that minimise induced drag.

Single-segment aileron designs have limitations on the obtainable roll-yaw control ratio because of their dependence on wing planform and ${B_3}$ . More complex aileron designs, like multi-segmented ailerons, with more degrees of freedom could also be used to control roll-yaw coupling, and is a topic of future work.