Vertical Tail Model

At a conceptual design level, the purpose of an aircraft’s vertical tail is two-fold. Firstly, it must provide stability in yaw. Secondly, it must provide adequate yaw control authority in critical flight conditions. For a multi-engine aircraft, the critical flight condition is typically an engine failure at low speeds. The vertical tail must be capable of providing sufficient sideforce in this case [Raymer, 1992]. The vertical tail must also provide adequate yaw rate acceleration during landing flare in crosswind conditions. The design of the vertical tail is therefore coupled to the size of the fuselage, the position of the engines, and the aircraft’s moment of inertia.

Model Assumptions

The high-level assumptions for this model are that the the horizontal tail is mounted on the fuselage, so as to not require a reinforced vertical tail structure, and that the aircraft has two engines.

Model Description

Variable tables are available for download below:

• Free variables
• Fixed variables

Vertical Tail Geometry and Structure

The variables that define geometry are illustrated in . The moment arm of the vertical tail is the distance from the aircraft to the aerodynamic center of the vertical tail, which is assumed to be at the quarter chord. The moment arm is therefore upper bounded by the distance from the to the leading edge of the tail at the root, the height of the mean aerodynamic chord above the fuselage, the sweep angle, and the mean aerodynamic chord.

\[l_{vt}\leq\Delta x_{lead_{vt}}+z_{\bar{c}_{vt}}{\tan(\Lambda_{LE})}+0.25\bar{c}_{vt} \label{eq:vtmomentarm}\]

The x-coordinates of the leading and trailing edge at the root are related by the root chord. The tail trailing edge is upper bounded by imposing a constraint that the tail root cannot extend beyond the end of the fuselage. Together these constraints put an upper bound on the moment arm of the tail based on the length of the fuselage.

\[\begin{split}\begin{aligned} {\Delta x_{trail_{vt}}} &\geq {\Delta x_{lead_{vt}}} + {c_{root_{vt}}} \label{eq:vtleading}\\ {l_{fuse}} &\geq {x_{CG}} + {\Delta x_{trail_{vt}}} \label{eq:vttrailing}\end{aligned}\end{split}\]

The vertical tail structure is sized by its maximum lift coefficient and the never-exceed speed.

\[\begin{aligned} L_{vt_{max}} &= \frac12 \rho_{TO} V_{ne}^2 S_{vt}C_{L_{v,max}}\end{aligned}\]

The remaining geometry and structural constraints were already introduced in the wing model. Constraints [eq:planformarea,eq:meanaerochord,eq:spanwisemac,eq:taperratio,eq:mintaperratio] are adapted to the vertical tail model to constrain its geometry, with two minor modifications. Constraint  can be relaxed from a signomial equality to a signomial inequality constraint, meanwhile Constraint needs to be implemented as a signomial equality constraint. The wing structure model from [Hoburg, 2013] is also reused, however, given that the vertical tail only has a half-span, the definitions of \(b_{vt}\), \(S_{vt}\), and \(W_{vt}\) differ from those of their wing counterparts.

Engine-out Condition

The first performance constraint specifies that the maximum moment exerted by the tail must be greater than or equal to the moment exerted by the engines in an engine-out condition, exacerbated by the windmill drag of the engine that is inoperative [Drela, 2011].

\[{L_{vt,EO}}{l_{vt}} \geq {D_{wm}} {y_{eng}} + {T_e} {y_{eng}}\]

The worst case engine out condition is likely to occur during takeoff, when the velocity is lowest but the engine force required to safely complete takeoff is highest. The force exerted by the vertical tail in this critical low speed case is constrained by its maximum lift coefficient, its reference area and the minimum dynamic pressure. As a conservative estimate, the \(V_1\) speed is used because it is the minimum speed after which a takeoff can be completed, following a critical engine failure.

\[{L_{vt,EO}} = \frac12{\rho_{TO}}{V_1}^{2} {S_{vt}} {C_{L_{vt,EO}}}\]

The 3D lift coefficient is constrained by the airfoil sectional lift coefficient using finite wing theory [Anderson, 2001].

\[C_{L_{vt,EO}}\left(1 + \frac{c_{l_{vt,EO}}}{\pi e_{vt} AR_{vt}}\right) \leq c_{l_{vt,EO}}\]

The windmill drag can, to a first approximation, be lower bounded using a drag coefficient and a reference area [Drela, 2011], in this case the area of the engine fan.

\[\begin{aligned} {D_{wm}} &\geq \frac12{\rho_{TO}}{V_1}^{2} {A_{fan}} {C_{D_{wm}}}\end{aligned}\]

Crosswind Landing Condition

The second performance constraint ensures the vertical tail can provide adequate yaw rate acceleration in a crosswind landing, where the moment of inertia was constrained at the system level (Section [chap:full_aircraft]). To provide a safety margin during cross-wind landing, \(C_{L_{vt,landing}}\) is taken to be 85% of takeoff \({C_{L_{vt}}}\).

\[\frac12{\rho_{TO}{V_{land}}^{2}} S_{vt} l_{vt} C_{L_{vt, landing}} \geq \frac{\dot{r}_{req}}{I_{z}}\]

Vertical Tail Drag

The vertical tail produces drag, regardless of the flight condition. Neglecting any induced drag, the parasitic drag coefficient, \(C_{D_{p_{vt}}}\), is set by a softmax affine fit of XFOIL[Drela, 1989]data for the symmetric NACA 0008 through 0020 airfoils. The fit considers airfoil thickness, Mach number, and Reynolds number. It was developed with GPfit and has an RMS error of 1.31%.

\[\begin{split}\begin{aligned} {D_{vt}} &\geq \frac12 {\rho_{\infty}} {V_\infty}^{2}{S_{vt}}{C_{D_{p_{vt}}}} \\ {C_{D_{p_{vt}}}}^{1.189} &\geq 2.44\times10^{-77} (Re_{vt})^{-0.528} (\tau_{vt})^{133.8} (M)^{1022.7} \\ &+ 0.003 (Re_{vt})^{-0.410} (\tau_{vt})^{1.22} (M)^{1.55} \nonumber\\ &+ 1.967\times10^{-4} (Re_{vt})^{0.214} (\tau_{vt})^{-0.04} (M)^{-0.14} \nonumber\\ &+ 6.590\times10^{-50} (Re_{vt})^{-0.498} (\tau_{vt})^{1.56} (M)^{-114.6} \nonumber\\ {Re_{vt}} &= \frac{ {\rho_\infty} {V_\infty} {\bar{c}_{vt}}}{{\mu}}\end{aligned}\end{split}\]