Lifting-Line Predictions for the Ideal Twist Effectiveness of Spanwise Continuous and Discrete Control Surfaces ^†

Featured Application

Preliminary design analysis of novel or traditional control surfaces/mechanisms used to minimize induced drag in multiple flight conditions.

Abstract

Modern materials and manufacturing technologies have allowed the construction of morphing wings that are able to continuously vary certain airfoil parameters such as twist, camber, or control surface deflection as a function of span. This work presents a twist effectiveness parameter as a means of comparing the ideal aerodynamic efficiency of spanwise continuous control surfaces (morphing wings) and spanwise discrete control surfaces (standard wings). A numerical algorithm is used to compute the twist effectiveness of both continuous and discrete control-surface designs over a wide range of planform shapes with evenly spaced actuation for inviscid, incompressible flow. Results included here show that using continuous control surfaces instead of discrete control surfaces reduces induced drag by less than 5% for most applications.

1. Introduction

Wing twist can be used to control the lift distribution on a wing. In general, twist can be categorized as either geometric or aerodynamic. Geometric twist is a spanwise variation in geometric angle of attack. A common form of geometric twist is washout, which is a twisting of the wing to reduce the angle of attack near the wing tips. Aerodynamic twist is a spanwise variation in zero-lift angle of attack. A common form of aerodynamic twist is a flap that extends over only a portion of the wing. In either scenario, wing twist always results in a spanwise variation in the section angle of attack relative to the section zero-lift angle of attack. Twist can be defined as

$$T\left(z\right)\equiv \alpha \left(z\right)-{\alpha }_{L0}\left(z\right)-{\left(\alpha -{\alpha }_{L0}\right)}_{\mathrm{root}}$$

(1)

where the components $\alpha \left(z\right)-{\alpha }_{\mathrm{root}}$ and $-{\alpha }_{L0}\left(z\right)+{\alpha }_{L0 \mathrm{root}}$ are the contributions from the local geometric and aerodynamic twist, respectively.

Any form of twist can be applied in a spanwise continuous or discontinuous manner. For example, washout is usually a continuous form of twist, while flaps are a discontinuous form of twist. The original Wright flyer used geometric continuous twist, referred to as wing warping, for roll control [1,2], but later models switched to discrete flaps for structural reasons. Most aircraft today use discontinuous twist in the form of flaps to control the lift distribution and resultant forces and moments, likely due to structural simplicity. However, with modern morphing technology, it is now possible to accurately control a continuous twist distribution (aerodynamic or geometric) during flight. For example, AFRL [3], NASA [4,5,6,7], and others [8,9,10,11,12,13] have developed various methods for varying the camber along a wing during flight, with an emphasis on maintaining continuity of twist along the span.

The main benefit claimed by those employing continuous twist control rather than discontinuous twist control is a reduction in drag over the flight envelope, and studies have reported varying levels of drag benefits [4,7,14,15,16,17,18,19,20,21,22,23]. In order to adequately quantify the drag benefits of real-time continuous twist, a consistent and rigorous methodology for drag comparison must be employed. Studies for comparing induced drag, parasitic drag, and total drag of various discrete and continuous designs must be conducted. Here, we provide one sample methodology and evaluate the induced-drag benefits of wings with continuous twist control compared to wings with discrete twist control.

Wings with mechanisms that control the twist distribution in a spanwise continuous manner are defined here as having a continuous control surface. In contrast, wings with mechanisms that vary the twist at discrete spanwise sections are defined here as having discrete control surfaces. Figure 1 visually indicates the difference between spanwise discrete and continuous control surfaces. Here, we use a numerical potential flow algorithm to evaluate ideal induced-drag solutions over a wide range of wing planforms and control mechanism designs. Comparisons are made between continuous and discrete control surfaces for comparable wing planforms and degrees of freedom within the respective control mechanisms. These results can aid in preliminary design of wings that use active control to minimize induced drag over the flight envelope. While the results presented here are limited to spanwise-symmetric planforms and twist distributions, and therefore spanwise-symmetric lift distributions, the method could be applied to asymmetric scenarios.

2. Materials and Methods

2.1. Theoretical Development

The prevailing theory that relates wing planform, twist, and angle of attack to the resulting lift and induced drag is the classical lifting-line theory published by Prandtl in 1918 [24,25]. Phillips presented a much more useful form of lifting-line theory which provides significant insight into these relationships [26,27,28,29]. The following development is an overview of that formulation. The solution to the fundamental lifting-line equation gives the vortex strength of the lifting line as a Fourier sine series:

$$\Gamma \left(\theta \right)=2b{V}_{\infty }{\displaystyle \sum }_{n=1}^N{A}_n\sin \left(n\theta \right), \theta \equiv {\cos }^{-1}\left(-{\displaystyle \frac{2z}{b}}\right)$$

(2)

where $\theta$ is a change in variables for the spanwise coordinate. The lift and induced-drag coefficients as functions of the Fourier coefficients in the absence of rolling rate are

$$C_L=\pi R_A{A}_1$$

(3)

$$C_{D_i}=\pi R_A{\displaystyle \sum }_{n=1}^{N}n{A}_n^2$$

(4)

In order to decouple the effects of angle of attack and twist, a change in variables can be used as suggested by Phillips [26,27,28]. A similar change in variables is used here, given by

$$A_n\equiv a_n{\left(\alpha -{\alpha }_{L0}\right)}_{\mathrm{root}}+b_n\Omega$$

(5)

where the decomposed Fourier coefficients $a_n$ and $b_n$ are related to the wing planform and twist distribution according to

$${{\displaystyle \sum }}_{n=1}^N{a}_n\left[{\displaystyle \frac{4b}{{\tilde{C}}_{L,\alpha }c\left(\theta \right)}}+{\displaystyle \frac{n}{\sin \left(\theta \right)}}\right]\sin \left(n\theta \right)=1$$

(6)

$${{\displaystyle \sum }}_{n=1}^N{b}_n\left[{\displaystyle \frac{4b}{{\tilde{C}}_{L,\alpha }c\left(\theta \right)}}+{\displaystyle \frac{n}{\sin \left(\theta \right)}}\right]\sin \left(n\theta \right)=\omega \left(\theta \right)$$

(7)

The term $\Omega$ is the amount of twist at the location of maximum twist magnitude of the wing and $\omega \left(\theta \right)$ is the twist distribution function normalized by $\Omega$. The twist distribution function $\omega \left(\theta \right)$ determines the shape of the twist distribution, while $\Omega$ determines the amount of twist. The twist distribution is then given as

$$T\left(\theta \right)=\Omega \omega \left(\theta \right)$$

(8)

Using Equations (5)–(7) in Equation (4) gives the induced-drag coefficient for a wing with an arbitrary $c\left(\theta \right)$, $\omega \left(\theta \right)$, and $\Omega$, as

$$C_{D_i}={\displaystyle \frac{C_L^2\left(1+{\kappa }_P\right)+{\kappa }_{PL}{C}_L{C}_{L.\alpha }\Omega +{\kappa }_{P\Omega }{\left(C_{L,\alpha }\Omega \right)}^2}{\pi R_A}}$$

(9)

where

$${\kappa }_P={\displaystyle \sum }_{n=2}^{N}n{\displaystyle \frac{a_n^2}{a_1^2}}$$

(10)

$${\kappa }_{PL}=2{\displaystyle \frac{b_1}{a_1}}{\displaystyle \sum }_{n=2}^{N}n{\displaystyle \frac{a_n}{a_1}}\left({\displaystyle \frac{b_n}{b_1}}-{\displaystyle \frac{a_n}{a_1}}\right)$$

(11)

$${\kappa }_{P\Omega }={\left({\displaystyle \frac{b_1}{a_1}}\right)}^2{\displaystyle \sum }_{n=2}^{N}n{\left({\displaystyle \frac{b_n}{b_1}}-{\displaystyle \frac{a_n}{a_1}}\right)}^2$$

(12)

For a given section lift slope, the term ${\kappa }_P$ is solely a function of $c\left(\theta \right)$, while the terms ${\kappa }_{PL}$ and ${\kappa }_{P\Omega }$ are functions of $c\left(\theta \right)$ and $\omega \left(\theta \right)$.

Lifting-line theory predicts that for a given wing and a given total lift coefficient, the least amount of induced-drag occurs when the lift distribution is elliptic. One way to achieve this is to have an elliptic planform shape with zero twist. An elliptic wing with no twist results in ${\kappa }_P=\Omega =0$, which from Equation (9) gives

$${\left(C_{D_i}\right)}_{\mathrm{elliptic}}={\displaystyle \frac{C_L^2}{\pi R_A}}$$

(13)

This is the absolute minimum induced drag a wing can produce for a given lift and wingspan [24,25,29]. Any other planform shape with zero twist will generate higher induced drag than that in Equation (13) for comparable lift coefficients and aspect ratios. Equation (13) can serve as a natural baseline for comparing the induced drag of other wing planforms and twist distributions. For a wing with an arbitrary $c\left(\theta \right)$ and $\Omega =0$, the induced-drag coefficient from Equation (9) reduces to

$$C_{D_i}={\displaystyle \frac{C_L^2}{\pi R_A}}\left(1+{\kappa }_P\right), {\kappa }_P={\displaystyle \frac{1}{e_s}}-1$$

(14)

where $e_s$ is commonly referred to as the span efficiency factor [27] and ${\kappa }_P$ is an induced-drag planform penalty. Examining Equations (13) and (14), we can see that ${\kappa }_P$ is the percent increase in induced drag that a wing without twist produces compared to the elliptic wing with the same aspect ratio at the same lift coefficient. Equations (6) and (10) show that ${\kappa }_P$ is a function of planform shape.

Figure 2 shows ${\kappa }_P$ for various linearly tapered planforms calculated using a numerical lifting-line code [30,31]. These solutions agree closely with solutions obtained from Equations (6) and (10). Similar results for the planform penalty factor have been presented before by Glauert [32], McCormick [33], and Phillips [26,27]. Glauert’s plot is the basis for the common rule of thumb that a taper ratio of about 0.5 will minimize induced drag. Results from Figure 2 show that, in the absence of twist, a taper ratio near 0.35 will minimize induced drag for most simply tapered wings over a wide range of aspect ratios. However, once twist is included, the optimum taper ratio can be significantly different.

It has been shown that wings with arbitrary planform can produce the elliptic lift distribution and therefore produce minimum induced drag by employing the optimal twist distribution given by

$${\omega }_{\mathrm{opt}}\left(\theta \right)=1-{\displaystyle \frac{\sin \left(\theta \right)}{c\left(\theta \right)/c_{\mathrm{root}}}}$$

(15)

$${\Omega }_{\mathrm{opt}}=-{\displaystyle \frac{4b{C}_L}{\pi R_A{\tilde{C}}_{L,\alpha }{c}_{\mathrm{root}}}}$$

(16)

$${\left(\alpha -{\alpha }_{L0}\right)}_{\mathrm{root}}={\displaystyle \frac{C_L}{\pi R_A}}\left({\displaystyle \frac{4b}{{\tilde{C}}_{L,\alpha }{c}_{\mathrm{root}}}}+1\right)$$

(17)

From Equations (15) and (16), we see that the optimal amount of twist depends on the lift coefficient, or flight condition, while the optimal twist distribution function is independent of operating condition and depends only on the planform shape. Since $\Omega$ depends on the flight condition, a wing with fixed twist is only optimal for one flight condition. At any other flight condition, the wing will produce a higher induced-drag coefficient than the optimum given in Equation (13). However, if a wing could continuously change the twist distribution during flight, the optimal twist distribution could be achieved for any flight condition and planform shape. The control surfaces on a wing can be used to minimize induced drag as much as possible for any given lift coefficient and planform shape. However, with a limited number of control surfaces, the absolute minimum induced drag given in Equation (13) may not be achievable. The ability of a given control-surface design to minimize drag will be called the twist effectiveness.

2.2. Twist Effectiveness

For a wing with a given $c\left(\theta \right)$, $\omega \left(\theta \right)$, and $C_L$, an optimal $\Omega$ can be found to minimize induced drag by taking the partial derivative of Equation (9) with respect to $\Omega$, setting the right-hand side equal to zero, and solving for the optimal $\Omega$. Using the optimal $\Omega$ in Equation (9) gives

$${\left(C_{D_i}\right)}_{\mathrm{opt}}={\displaystyle \frac{C_L^2}{\pi R_A}}\left(1+{\kappa }_{Do}\right),\hspace{1em}\hspace{1em}{\kappa }_{Do}={\kappa }_P-{\displaystyle \frac{{\kappa }_{PL}^2}{4{\kappa }_{P\Omega }}}$$

(18)

The parameter ${\kappa }_{Do}$ is the optimum induced-drag factor and shows the minimum induced drag possible for a given planform and twist distribution function. It is similar to ${\kappa }_P$ in that it is the percent increase in drag from the absolute minimum induced drag given in Equation (13) to that of a wing with a given planform, twist distribution function, and an optimum amount of twist. In other words,

$${\kappa }_{Do}={\displaystyle \frac{{\left(C_{D_i}\right)}_{\mathrm{opt}}-{\left(C_{D_i}\right)}_{\mathrm{elliptic}}}{{\left(C_{D_i}\right)}_{\mathrm{elliptic}}}}$$

(19)

The term ${\kappa }_{Do}$ is a function of both $c\left(\theta \right)$ and $\omega \left(\theta \right)$. If both of these are known, ${\kappa }_{Do}$ can be calculated from Equations (10)–(12) and (18). Note that up to this point, no constraint has been applied to the twist distribution function and $\omega \left(\theta \right)$ is completely arbitrary. In order to compare the induced-drag performance of spanwise discrete and continuous control surfaces, the best twist distribution function possible should be used for each case. The best twist distribution function a wing can have is given in Equation (15), but this would require infinitely fine spanwise control. Realistic designs are limited by constraints on the twist distribution function and twist amount such that the twist distribution is always within the capabilities of the control mechanism. Determining an optimal twist distribution function for a given control mechanism can be challenging. Towards this goal, ${\kappa }_{Do}$ can be redefined into a more useful form by

$${\kappa }_{Do}\equiv {\kappa }_P\left(1-{\epsilon }_T\right),\hspace{1em}\hspace{1em}{\epsilon }_T={\displaystyle \frac{{\kappa }_{PL}^2}{4{\kappa }_{P\Omega }{\kappa }_P}}$$

(20)

where ${\epsilon }_T$ is the twist effectiveness. The twist effectiveness, ${\epsilon }_T$, is a measure of how well the twist distribution corrects for ${\kappa }_P$. The twist effectiveness can vary from 0.0 to 1.0, with 0.0 meaning that the twist cannot correct for any of the planform penalty, and 1.0 meaning that the twist can completely correct for the planform penalty.

For example, if we had a wing with a taper ratio of 1 and an aspect ratio of 18, the induced-drag planform penalty would be roughly 0.15, see Figure 2. This means that the wing without any twist will produce roughly 15% more induced drag than the optimum. However, if the twist distribution given in Equations (15) and (16) is used, ${\epsilon }_T=1$, and the control mechanism is capable of correcting for 100% of the planform penalty. If instead, the best the control mechanism was able to achieve was a twist effectiveness of 0.5, then the control mechanism would only correct for 50% of the planform penalty and the wing would produce roughly 7.5% more than the optimum, ${\kappa }_{Do}$ = 0.075. The twist effectiveness term by itself is not enough information to determine actual changes in induced drag since it is the measure of how well the twist distribution corrects for the induced-drag planform penalty. Using Equation (20) in Equation (18) allows the induced-drag coefficient to be rewritten as

$${\left(C_{D_i}\right)}_{\mathrm{opt}}={\displaystyle \frac{C_L^2}{\pi R_A}}\left[1+{\kappa }_P\left(1-{\epsilon }_T\right)\right]$$

(21)

This formulation combines all the components of ${\kappa }_{Do}$ dependent on twist into one parameter, ${\epsilon }_T$, and helps in understanding and comparing the effectiveness of different control mechanisms.

2.3. Numerical Solver

For any given $c\left(\theta \right)$ and control mechanism, ${\epsilon }_T$ can be computed with the aid of aerodynamic tools and numerical optimization algorithms. Here, we use MachUp, a modern numerical lifting-line algorithm to predict the aerodynamics for a given planform and control configuration [30,31]. MachUp differs from what is commonly referred to as the numerical lifting-line algorithm [34] in that it solves for the vorticity strength at each wing section by relating the lift predicted from the section airfoil properties to the lift predicted by the three-dimensional vortex-lifting law [35], instead of enforcing a Neumann condition at the three-quarter-chord location [34]. In this regard, MachUp is more analogous of a numerical lifting-line method than that from Katz and Plotkin [34] since it uses the same fundamental relation as was used by Prandtl in classical lifting-line theory [24,25]. MachUp extends past the applicable scope of classical lifting-line by being able to model multiple lifting surfaces, each with arbitrary sweep and dihedral, even though this work considers only single wings with no sweep or dihedral. This method has shown good agreement with computational fluid dynamics solutions and experimental results for wings with aspect ratios greater than about 4 [30].

MachUp is used in conjunction with the open-source Sequential Least Squares Programming optimization algorithm [36]. The optimization algorithm calls MachUp as its objective function to determine the induced drag of the wing and allows the twist deflections to vary at the control actuation locations to minimize induced drag and maintain a given lift coefficient. For a given wing, the optimization algorithm runs multiple times for $C_L$ values 0.5, 0.8, 1.1, and 1.4 to give a range of drag versus lift data. Then, using the ${\kappa }_P$ values from Figure 2, Equation (21) is fit to the optimization data by varying ${\epsilon }_T$ such that the root mean squared (RMS) of the percent error is minimized. This provides an accurate estimate of ${\epsilon }_T$ for the given wing planform and control design. For all solutions shown here, a section lift slope of $2\pi$ was used, and an RMS error less than 1.35 × 10⁻⁶ was obtained. Since the airfoil properties are modeled as linear functions without stall and since $C_{D_i}$ is exactly proportional to $C_L^2$ according to lifting-line theory, the choice of $C_L$ values for this process is somewhat arbitrary. It was found that the optimization algorithm could find solutions more effectively at higher lift coefficients where gradients are stronger. Therefore, the high lift coefficient values used here are merely a means to compute a more accurate twist effectiveness term at low angles of attack. The choice of this range of lift coefficients does not imply that anything in this work applies near or beyond stall conditions.

The analysis was only applied to a specific class of control mechanisms, even though it could be applied to others. For this class of control mechanisms, the actuators were assumed to be spaced evenly in the spanwise direction. The continuous control surfaces are modeled by having a linear variation in geometric twist between control actuators, with one always at the wing root and another always at the wing tip. Since twist is defined relative to the root, the actuator at the root determines the angle of attack of the wing. The discrete control surfaces are modeled by equally sized discrete sections of constant geometric twist. The number of sections is determined by the number of control actuators per semispan, where again the actuator at the root determines the angle of attack of the wing. Figure 1 shows an example of a four-actuator discrete and continuous control mechanism using trailing-edge flaps instead of geometric twist for clarity. The actuator deflections are equivalent between the two mechanisms shown.

This class of control mechanisms has no limits placed on the magnitudes of the control deflections. From Equation (16), we expect the magnitude of control deflections to increase with increasing lift coefficient. For a control mechanism with deflection limits, a lift coefficient will eventually be reached that saturates one or more of the control deflections from the mechanism. The twist effectiveness term is constant in the range of lift coefficients from 0 up to this saturating lift coefficient. Above this saturating lift coefficient, the twist effectiveness would vary with lift coefficient and would likely decrease. The results presented here show the cases when the control mechanism has not saturated.

3. Results

Using the process described above, ${\epsilon }_T$ can be calculated for any planform and control mechanism. This process was repeated over the range $0\le R_T\le 1$ and $4\le R_A\le 20$. Results are shown in Figure 3 for wings with 2–5 actuators per semispan. The reader is reminded that the twist effectiveness ${\epsilon }_T$ is a measure of how well the control mechanism can negate the drag increase caused by the planform shape. A twist effectiveness of 1.0 means that the control mechanism is 100% capable and the wing can operate at the lowest induced drag possible for that particular aspect ratio, while a twist effectiveness of 0.0 means that the control mechanism is not capable of reducing the induced drag for that wing geometry.

Notice that for both discrete and continuous control surfaces, as the number of actuators per semispan increases, the twist effectiveness improves. Recall that the twist effectiveness was defined as using the best twist distribution possible given the limitations of the control mechanism. Since a control mechanism with more degrees of freedom is better able to match the optimal solution given in Equations (15) and (16), it is expected that the twist effectiveness improves with increasing number of actuators. Also note for any given number of actuators, the continuous control surface generally has better twist effectiveness than the discrete control surface. One exception to this is the two-actuator case in the region of approximately 0.35–0.4 taper ratio, where the discrete control surface has a very slight advantage. This range of taper ratios also appears to be a minimum-value region for the twist effectiveness for the 2–5-actuator cases. This is interesting, especially when considering Figure 2, because the taper ratio that provides the best ${\kappa }_P$provides the worst ${\epsilon }_T$. One potential explanation for this is that the planform penalty is already very low, so there is less the control mechanism can do to minimize induced drag.

It is not uncommon to use a taper ratio near 0.5 for wings that do not use active twist to minimize induced drag. However, in light of Figure 3, if wing twist with only a few actuators is to be used to minimize induced drag, a taper ratio near 0.5 is extremely ineffective. Clearly, both the planform and control-mechanism design should be taken into account when designing a wing that actively uses the control mechanism to minimize drag.

From Equation (20), we see that ${\kappa }_P$ and ${\epsilon }_T$ combine to give the optimum induced-drag factor, ${\kappa }_{Do}$. Results for ${\kappa }_{Do}$ are shown in Figure 4 for different planforms and control mechanisms. Results similar to the continuous cases in Figure 4a have been presented by Phillips [26,27] in the context of a linearly tapered wing with optimal linear washout. Note that the vertical scale between the subplots in Figure 4 decreases significantly with increasing number of actuators. As expected, the more actuators used, the more closely it can produce an elliptic lift distribution. Also note that, as expected, the continuous control surfaces can nearly always outperform the discrete control surfaces for a given number of actuators, except for the region in the two-actuator case discussed previously. However, the question arises whether the continuous control surfaces provide sufficient benefit over discrete control surfaces to offset the likely increase in manufacturing complexity and weight.

It is also insightful to compare the performance of the continuous and discrete control surfaces for a given planform in terms of a percent difference in minimum induced drag between the two designs. Using the discrete control surface case as a baseline, this can be found from

$$\begin{gathered} \Delta {\left(C_{D_i}\right)}_{\mathrm{opt}}\equiv {\displaystyle \frac{{\left(C_{D_i \mathrm{cont}}\right)}_{\mathrm{opt}}-{\left(C_{D_i disc}\right)}_{\mathrm{opt}}}{{\left(C_{D_i \mathrm{disc}}\right)}_{\mathrm{opt}}}}={\displaystyle \frac{{\kappa }_{Do \mathrm{cont}}-{\kappa }_{Do disc}}{1+{\kappa }_{Do disc}}} \\ ={\displaystyle \frac{{\kappa }_P\left({\epsilon }_{T disc}-{\epsilon }_{T \mathrm{cont}}\right)}{1+{\kappa }_{Do \mathrm{disc}}}} \end{gathered}$$

(22)

Results for Equation (22) are shown in Figure 5. From Figure 5, we see that a continuous control-surface design will reduce induced drag by less than 5% relative to an equivalent planform with discrete control surfaces for nearly all cases. Also note that the difference in drag between the two control surfaces diminishes with increasing number of actuators with less than 3% benefit for the three-actuator cases, less than 2% benefit for the four-actuator cases, and less than about 1% for the five-actuator cases. This is because there is an absolute minimum induced drag possible for any given aspect ratio, and as the number of actuators increase, both control-mechanism designs are better able to match the twist distribution given by Equations (15) and (16).

4. Discussion

A theoretical development based on lifting-line theory is presented which gives three methods for comparing the ability of continuous and discrete control surfaces to minimize induced drag. These include the twist effectiveness ${\epsilon }_T$, the optimal induced-drag factor ${\kappa }_{Do}$, and a percent change in induced drag $\Delta {\left(C_{D_i}\right)}_{\mathrm{opt}}$. A numerical process is also presented which is capable of calculating these parameters for any given planform shape and control mechanism. This methodology was used to study discrete and continuous twist distributions on wings with evenly spaced actuators.

Wings without twist experience an induced-drag planform penalty when the planform is not elliptic. This induced-drag planform penalty is shown in Figure 2. The twist effectiveness ${\epsilon }_T$ is a measure of how well the wing’s control mechanism is able to negate the induced-drag planform penalty for any given flight condition and is shown in Figure 3 for discrete and continuous control surfaces with varying number of actuators per semispan. These plots show that the continuous control surfaces generally perform better than discrete control surfaces. These results also show that, in general, control mechanisms are less effective at minimizing induced drag in the taper ratio range of 0.3–0.4. This is insightful, since this is near the taper ratio range that gives the smallest induced-drag planform penalty. Therefore, both the wing planform and control-mechanism design should be taken into account when designing a wing that uses the control mechanism to actively reduce induced drag. Figure 3 also shows that as the number of actuators within the control mechanism increase, twist effectiveness improves. This is because higher degrees of freedom within the control mechanism, allow it to better match the optimal twist distribution given in Equations (15) and (16).

The optimal induced-drag factor ${\kappa }_{Do}$ combines the effects of planform and control-mechanism design and is given in Equation (20). This allows for both the induced-drag planform penalty and twist effectiveness to be considered when designing a wing. Figure 4 shows the optimal induced-drag factor which is the optimal induced drag possible for a given wing planform and control-mechanism design relative to the absolute minimum induced drag for the same aspect ratio. When viewed from this perspective, the continuous control surfaces again noticeably outperform discrete control surfaces. These results also show that the desired taper ratio is highly dependent on the control mechanism design, and that discrete control surfaces are most effective with taper ratios in the range of 0.1–0.2, while continuous control surfaces are most effective at either relatively high or low taper ratios.

The percent difference in induced drag between using a continuous and discrete control surface $\Delta {\left(C_{D_i}\right)}_{\mathrm{opt}}$ can be helpful to determine whether there is sufficient aerodynamic benefit to offset the likely increase in manufacturing complexity and weight for using the continuous control surface. This percent difference is given in Equation (22), and results are shown in Figure 5. From the best case of the results presented here, wings with a continuous control surface can have an induced-drag reduction of just over 5% compared to an equivalent wing with discrete control surfaces. The difference in drag between the two control surfaces diminishes with increasing number of actuators with less than 3% benefit for the three-actuator cases, less than 2% benefit for the four-actuator cases, and less than about 1% for the five-actuator cases. This is caused by the fact that higher degrees of freedom allow the control mechanism, whether discrete or continuous, to approach the optimal solution.

The dependence of the performance of an aircraft on drag is complex. Therefore, a blanket statement about how a reduction in drag would affect the overall performance of aircraft is difficult to make. However, it is insightful that a 5% reduction in induced drag can roughly be correlated with a 2.5% reduction in thrust, if induced drag was originally 50% of the total drag. This could translate to about a 2.5% reduction in fuel required, which would lower the weight of the aircraft and reciprocally result in further drag reduction.

Continuous control surfaces, in general, have greater aerodynamic efficiency capabilities than discrete control surfaces. However, the ideal aerodynamic benefits for induced drag shown in this study are small. It should be noted that lifting-line theory used here to determine the ideal aerodynamics does not take into account any flow spillage that may occur across the gap created by two adjacent discrete control surfaces that have different deflections. Accounting for this would likely cause the discrete control surfaces to have more drag than shown here. Future work could include comparing efficiencies of spanwise discrete and continuous control surfaces while taking into account parasitic drag and the discrete control surface flow spillage effect.

As a final note, the present manuscript includes two main contributions. The first is simply a method for comparing the effectiveness of discrete control surfaces to continuous. The method followed here could be repeated using wind-tunnel experiments, computational fluid dynamics, or higher-fidelity aerodynamic prediction methods. The second contribution is the presentation of results of this method based on lifting-line theory. The limitations of lifting-line theory are well documented in the literature. As a reminder, the results shown here neglect influence of viscosity, and should not be expected to be accurate for wings with aspect ratios less than about 4. Additionally, compressibility effects have been neglected. Nevertheless, the trends obtained from these results are insightful and can guide the discussion and development of future morphing-wing controls.