Wing Model

The aircraft wing is designed generate sufficient lift such that the aircraft can takeoff, climb, cruise, descend, and land safely. Typically the wings also carry fuel tanks and support the engines. Unfortunately, wings are heavy and produced drag. The purpose of this model is to relate all of these considerations.

Model Assumptions

The wing model assumes a continuous-taper, low-wing configuration with a modern transonic airfoil. It does not currently consider wing twist or wing dihedral. It also does not consider roll or yaw stability.

Model Description

Variable tables are available for download below:

• Free variables
• Fixed variables

Wing Geometry

Before considering a wing’s performance, the variables that prescribe its geometry must be appropriately constrained.

The relationship between reference area, span and mean geometric chord is enforced using a constraint that assumes a trapezoidal planform. This constraint is implemented as a signomial equality constraint because there is both upward and downward (optimization) pressure on the reference area, and it is not possible to know a priori which will dominate.

\[{S_{w}} = {b_{w}} \frac{c_{root_{w}} + c_{tip_{w}}}{2} \label{eq:planformarea}\]

The mean aerodynamic chord relationship for a trapezoidal wing can be written as a signomial constraint, and its spanwise location can be written as a monomial equality constraint. These constraints make use of dummy variables, \(p_w\) and \(q_w\), introduced by the structural model below.

\[\begin{split}\begin{aligned} \bar{c}_{w} &\leq \frac23 \left(\frac{1 + \lambda_{w} + \lambda_{w}^2}{q_{w}}\right) c_{root_{w}} \label{eq:meanaerochord} \\ y_{\bar{c}_w} &= \frac{b_w q_w}{3 p_w} \label{eq:spanwisemac}\end{aligned}\end{split}\]

The wing taper ratio is defined by a monomial equality constraint. It is necessary to lower bound taper to avoid an unacceptably small Reynolds number at the wing tip [Kroo, 2001]. For the purpose of this work, the taper is lower bounded using the taper ratio of the reference aircraft’s wing.

\[\begin{split}\begin{aligned} \lambda_{w} &= \frac{c_{tip_{w}}}{c_{root_{w}}} \label{eq:taperratio}\\ {\lambda_{w}} &\geq \lambda_{w_{min}} \label{eq:mintaperratio}\end{aligned}\end{split}\]

Finally, a maximum span constraint can be imposed to reflect, for example, a gate size constraint.

\[b_w \leq b_{w,max}\]

Wing Lift

Total lift is constrained to be greater than the weight of the aircraft plus the downforce from the horizontal tail. The constant \(f_{L_{total/wing}}\) is greater than one and used to account for fuselage lift.

\[\begin{split}\begin{aligned} L_{total} &\geq W + L_{ht}\\ L_{total} &= f_{L_{total/wing}} L_{w}\end{aligned}\end{split}\]

The standard equation for the lift of a wing is a natural monomial equality constraint.

\[\begin{aligned} L_w = \frac12 \rho_{\infty} V_{\infty}^2 S_w C_{L_w}\end{aligned}\]

However, this assumes a continuous unobstructed wing planform. Correcting for lift loss at the fuselage and at the wing tips, gives the adjusted Equation , which can be rearranged into the posynomial Constraint .

\[\begin{split}\begin{aligned} L_w &= \frac12 \rho_{\infty} V_{\infty}^2 S_w C_{L_w} - \Delta L_o - 2 \Delta L_t \label{eq:liftadjeq} \\ \frac12 \rho_{\infty} V_{\infty}^2 S_w C_{L_w} &\geq L_w + \Delta L_o + 2 \Delta L_t \label{eq:liftadjcon}\end{aligned}\end{split}\]

The lift corrections are given as monomial equality constraints [Drela, 2011].

\[\begin{split}\begin{aligned} \Delta L_o &= \eta_o f_{L_o} \frac{b_w}{2} p_o \\ \Delta L_t &= f_{L_t} p_o c_{root_{w}} \lambda_w^2\end{aligned}\end{split}\]

The lift coefficient of the wing goes linearly with the angle of attack, which is limited by a maximum angle of attack due to stall.

\[\begin{split}\begin{aligned} C_{L_w} &= C_{L_{\alpha,w}}\alpha_w \\ \alpha_{w} &\leq \alpha_{w,max}\end{aligned}\end{split}\]

The DATCOM formula is an analytic function for estimating the lift curve slope of a wing or tail, based on empirical results [Kroo, 2001].

\[C_{L_{\alpha,w}} = \frac{2 \pi AR_{w}}{2+\sqrt{(AR_{w}/\eta_w)^2(1+\tan^2\Lambda - M^2)+4}}\]

This relationship can be used as a signomial inequality to constrain the lift curve slope, although some algebraic manipulation is needed.

\[\begin{split}\begin{aligned} C_{L_{\alpha,w}} &\leq \frac{2 \pi AR_{w}}{2+\sqrt{(AR_{w}/\eta_w)^2(1+\tan^2\Lambda-M^2)+4}} \\ (AR_{w}/\eta_w)^2(1+\tan^2\Lambda - M^2)+4 &\leq \left( \frac{2\pi AR_{w}}{C_{L_{\alpha,w}}} - 2 \right)^2 \\ (AR_{w}/\eta_w)^2(1+\tan^2\Lambda - M^2) &\leq \frac{4 \pi^2 AR_{w}^2}{C_{L_{\alpha,w}}^2} - \frac{8 \pi AR_{w}}{C_{L_{\alpha,w}}} \\ \frac{C_{L_{\alpha,w}}^2}{\eta_w^2}\left(1 + \tan^2\Lambda - M^2\right) + \frac{8\pi C_{L_{\alpha,w}}}{AR_{w}} &\leq 4\pi^2 \end{aligned}\end{split}\]

Maximum wing lift is constrained using an assumed load factor, \(N_{lift}\).

\[\label{e:Lmax} f_{L_{total/wing}} L_{w_{max}} \geq N_{lift} W_{total} + L_{ht_{max}}\]

Finally, wing loading is constrained to be less than a user specified maximum.

\[\begin{split}\begin{aligned} W_{S} &= \frac{1}{2} \rho_{\infty} C_{L_w} {V_{\infty}}^2 \\ W_{S} &\leq W_{S_{max}}\end{aligned}\end{split}\]

Wing Weight

Wing weight is constrained to be greater than the wing structural weight plus a series of fractional weights to account for wing ribs and control surfaces.

\[W_{wing} \geq W_{struct_{w}}(1 + f_{flap} + f_{slat} + f_{aileron} + f_{lete} + f_{ribs} + f_{spoiler} + f_{watt})\]

Wing structural weight is constrained using an adaptation of the structural model from [Hoburg, 2013], which comprises 12 monomial and posynomial constraints.

\[\begin{split}\begin{aligned} {W_{struct_{w}}} &\geq ({W_{cap}} + {W_{web}}) \\ {W_{cap}} &\geq \frac{8{\rho_{cap}} {g} {w} {t_{cap}} {S_{w}}^{1.5} {\nu}}{3{AR_{w}}^{0.5}} \\ {W_{web}} &\geq \frac{8{\rho_{web}}{g}{r_h}{\tau_{w}}{t_{web}}{S_{w}}^{1.5}{\nu}}{3{AR_{w}}^{0.5}} \\ {\nu}^{3.94} &\geq 0.14{p_{w}}^{0.56} + \frac{0.86}{{p_{w}}^{2.4}} \\ {p_{w}} &\geq 1 + 2{\lambda_{w}} \\ 2{q_{w}} &\geq 1 + {p_{w}} \\ \frac{0.92^2}{2}{\tau_{w}}^{2}{t_{cap}}{w} &\geq 0.92{\tau_{w}}{t_{cap}}^{2}{w} + {I_{cap}} \\ \frac{{AR_{w}} {M_r} {N_{lift}} {\tau_{w}} {q_{w}}^{2}}{{I_{cap}} {S_{w}} {\sigma_{max}}} &\leq 8 \\ \frac{{AR_{w}}{L_{w_{max}}}{N_{lift}}{q_{w}}^{2}}{{S_{w}}{\sigma_{max,shear}}{\tau_{w}}{t_{web}}}&\leq 12 \\ {AR_{w}} &= \frac{{b_{w}}^{2}}{{S_{w}}} \\ {\tau_{w}} &\leq 0.14 \end{aligned}\end{split}\]

The original root bending moment constraint,

\[{M_r} \geq \frac{{AR_{w}} {L_{w_{max}}} {p_{w}}}{24},\]

is replaced with a more sophisticated signomial constraint that considers the load relief effect due to the weight of the engine and the fuel tanks. To derive the constraint, the lift per unit span of wing is assumed to be proportional to the local chord, and the wing planform area is partitioned into an untapered (rectangular) area \(A_{rect}\) and a fully tapered (triangular) area \(A_{tri}\).

\[\begin{split}\begin{aligned} A_{tri} &= \frac{1}{2} (1-\lambda_w) c_{root_{w}} b_w \\ A_{rect} &= c_{tip_{w}} b_w\end{aligned}\end{split}\]

The wing area component loads are treated as point loads to determine the equivalent wing root moment.

\[\begin{split}\begin{aligned} \label{eq:M_rinit} M_r c_{root_{w}} \geq &\left(L_{w_{max}} - N_{lift}\left(W_{wing} + f_{fuel,wing} W_{fuel}\right)\right) \left(\frac16 A_{tri} + \frac14 A_{rect}\right)\frac{b_{w}}{S_{w}} \\ &- N_{lift} W_{engine} y_{eng} \nonumber\end{aligned}\end{split}\]

This constraint can be further simplified to remove the need for intermediary variables \(A_{tri}\) and \(A_{rect}\), since

\[\begin{split}\begin{aligned} \frac{1}{6} A_{tri} + \frac{1}{4} A_{rect} &= \frac{1}{12} (c_{root_{w}} - c_{tip_{w}}) b_{w} + \frac{1}{4} c_{tip_{w}} b_{w} \\ &= \frac{b_{w}}{12} (c_{root_{w}} + 2 c_{tip_{w}}). \label{eq:Asub}\end{aligned}\end{split}\]

Substituting Equation  into Constraint  yields the following wing root moment constraint.

\[\begin{split}\begin{aligned} M_r c_{root_{w}} \geq &\left(L_{w_{max}} - N_{lift}\left(W_{wing} + f_{fuel,wing} W_{fuel}\right)\right) \left(\frac{b_{w}^2}{12 S_{w}} \left(c_{root_{w}} + 2 c_{tip_{w}}\right)\right) \\ & - N_{lift} W_{engine} y_{eng} \nonumber\end{aligned}\end{split}\]

Note that this provides a conservative estimate for the root moment, since it assumes that the lift per unit area is constant throughout the wing, whereas in reality the lift per unit area diminishes towards the wingtips.

Wing Drag

Wing drag is captured by five monomial and posynomial constraints. The parasitic drag coefficient is constrained using a softmax affine fit of XFOIL[Drela, 1989]simulation data for the TASOPT[Drela, 2011] C-series airfoils, which are representative of modern transonic airfoils[Drela, 2011]. The fit, which considers wing thickness, lift coefficient, Reynolds number, and Mach number, was developed with GPfit and has an RMS error of approximately 5%. Constraint  is an adaption of the standard definition of the induced drag coefficient [Anderson, 2001], with an adjustment factor for wingtip devices.

\[\begin{split}\begin{aligned} D_w &= \frac12 \rho_{\infty} V_{\infty}^2 S_w C_{D_w} \label{eq:wingdrag}\\ C_{D_w} &\geq C_{D_{p_w}} + C_{D_{i_w}} \label{eq:wingdragcoeff}\\ \label{eq:wingpdragcoeff} C_{D_{p_w}}^{1.65} &\geq 1.61 \left(\frac{Re_w}{1000}\right)^{-0.550} (\tau_{w})^{1.29} (M \cos(\Lambda))^{3.04} C_{L_w}^{1.78} \\ &+ 0.0466 \left(\frac{Re_w}{1000}\right)^{-0.389} (\tau_{w})^{0.784} (M \cos(\Lambda))^{-0.340} C_{L_w}^{0.951} \nonumber \\ &+ 191 \left(\frac{Re_w}{1000}\right)^{-0.219} (\tau_{w})^{3.95} (M\cos(\Lambda))^{19.3} C_{L_w}^{1.15} \nonumber \\ &+ 2.82e-12 \left(\frac{Re_w}{1000}\right)^{1.18} (\tau_{w})^{-1.76} (M \cos(\Lambda))^{0.105} C_{L_w}^{-1.44} \nonumber \\ \label{eq:wingRe} Re_w &= \frac{\rho_{\infty} V_{\infty} \bar{c}_w}{\mu} \\ C_{D_{i_w}} &\geq f_{tip} \frac{C_{L_w}^2}{\pi e AR_{w}} \label{eq:induceddrag}\end{aligned}\end{split}\]

The Oswald efficiency is constrained by a relationship from [Nita, 2012], in which the authors fit a polynomial function to empirical data. Given that all polynomials are signomials, this can easily be used in the framework.

\[e\leq \frac{1}{1 + f(\lambda_w) AR_{w} }\]

\[\label{eq:flambda} f(\lambda_w) \geq 0.0524 \lambda_w^4 - 0.15 \lambda_w^3 + 0.1659 \lambda_w^2 - 0.0706 \lambda_w + 0.0119\]

The Oswald efficiency is plotted as a function of taper ratio, as imposed by this pair of constraints, in .

Wing Aerodynamic Center

The true aerodynamic center and the of the wing are shifted in the aircraft’s x-axis with respect to the wing root quarter chord due to the swept geometry of the wing. This effect is captured with the variable \(\Delta x_{ac_w}\). Assuming that the wing lift per unit area is constant, and by integrating the product of the local quarter chord offset \(\delta x(y)\) and local chord area \(c(y)dy\) over the wing-half span, it can be calculated by

\[\label{eq:dXACwingDerivation} \Delta x_{ac_w} = \frac{2}{S} \int_{0}^{b/2} c(y) \delta x(y) dy,\]

where the local root chord \(c(y)\) and the local quarter chord offset \(\delta x(y)\) are given by:

\[\begin{split}\begin{aligned} \label{eq:cy} c(y) &= \left(1 - (1-\lambda_w) \frac{2y}{b_w} \right) c_{root_{w}} \\ \label{eq:dxy} \delta x(y) &= y \tan(\Lambda)\end{aligned}\end{split}\]

By substituting Equations and into Equation , expanding out the integral and relaxing the equality, \(\Delta x_{ac_w}\) can be constrained as follows.

\[\Delta x_{ac_w} \geq \frac{1}{4} \tan(\Lambda) AR_{w} c_{root_{w}} \left(\frac{1}{3} + \frac{2}{3} \lambda_w \right)\]

Fuel Volume

Fuel tanks are typically located inside the wingbox. Using the geometry of a TASOPT-optimized 737-800[Drela, 2011], a constraint on the maximum fuel volume in the wing was developed. For a wing of the same mean aerodynamic chord, thickness, and span as a TASOPT 737-800, the maximum available fuel volumes in the wing will match exactly. To allow for the possibility of auxiliary tanks in the horizontal tail or fuselage the user-specified value \(f_{fuel, usable}\) is introduced.

\[\begin{split}\begin{aligned} \label{e:V_fuel} V_{fuel, max} &\leq 0.303 {\bar{c}_w}^2 b_{w} \tau_{w} \\ W_{fuel_{wing}} &\leq \rho_{fuel} V_{fuel, max} g \\ W_{fuel_{wing}} &\geq \frac{f_{fuel, wing} W_{fuel_{total}}}{ f_{fuel, usable}}\end{aligned}\end{split}\]