Drag

The viscous calculation produces displacement, momentum, and kinetic energy areas $\Delta^*, \Theta, \Theta^* {\scriptstyle (x)}$.

The cross-sectional area over the center cylindrical portion is $A_{\rm fuse}$, which has already been defined by

\[\begin{aligned} A_{\rm fuse} = \left[ \pi + n_{\rm fweb}\left( 2\theta_{\rm fb}+ \sin 2 \theta_{\rm fb}\right) \right] R_{\rm fuse}^2 \;+\; 2 \left[ R_{\rm fuse}+ n_{\rm fweb}w_{\rm fb}\right] \Delta R_{\rm fuse}. \end{aligned}\]

This also defines the radius of the equivalent round cylinder. $\begin{aligned} R_{\rm cyl}& = & \sqrt{\frac{A_{\rm fuse}}{\pi}} \end{aligned}.$

The equivalent radii over the tapering nose and radius are then defined via the following convenient functions.

\[\begin{aligned} R {\scriptstyle (x)}& = & \left\{ \begin{array}{lcl} \displaystyle R_{\rm cyl} \left[ \: 1 - \left( \frac{x_{{\rm blend}_1} \!-\! x}{x_{{\rm blend}_1} \!-\! x_{\rm nose}} \right)^{\!\! a} \; \right]^{\! 1/a} & , & x_{\rm nose}< x < x_{{\rm blend}_1} \\[1.0em] \displaystyle R_{\rm cyl} & , & x_{{\rm blend}_1} < x < x_{{\rm blend}_{\,2}} \\[0.5em] \displaystyle R_{\rm cyl} \left[ \: 1 - \left( \frac{x \!-\! x_{{\rm blend}_{\,2}}}{x_{\rm end} \!-\! x_{{\rm blend}_{\, 2}}} \right)^{\!\! b} \; \right] & , & x_{{\rm blend}_{\,2}} < x < x_{\rm tail} \end{array} \right. \\ a & \simeq 1.6 & \\ b & \simeq 2.0 & \end{aligned}\]

The $x_{{\rm blend}_1}$ and $x_{{\rm blend}_{\,2}}$ locations are the nose and tailcone blend points, and do not necessarily have to be exactly the same as the $x_{{\rm shell}_1}$ and $x_{{\rm shell}_{\,2}}$ locations which define the loaded pressure shell. Likewise, $x_{\rm end}$ is the aerodynamic endpoint of the tailcone, and is distinct from its structural endpoint $x_{\rm conend}$. The $a$ and $b$ constant values above give reasonable typical fuselage shapes.

If the fuselage is nearly round, the necessary area and perimeter distributions follow immediately.

\[\begin{aligned} A {\scriptstyle (x)}& = & \pi \, {R{\scriptstyle (x)}}^2 \\ b_0 {\scriptstyle (x)}& = & 2 \pi R {\scriptstyle (x)} \end{aligned}\]

This would be suitably modified for non-circular cross-sections.

With this geometry definition, the viscous/inviscid calculation procedure provides the momentum and kinetic energy area distributions along the body and wake,

\[\begin{aligned} \left\{ \Theta {\scriptstyle (s)}\, , \: \Theta^* {\scriptstyle (s)}\right\} & = & f_{\rm f_{excr}} \: {\cal F}(M_{{\scriptscriptstyle \infty}}, Re_\ell\, ; \, A {\scriptstyle (x)}, b_0 {\scriptstyle (x)}) \end{aligned}\]

where ${\cal F}$ denotes the overall viscous/inviscid calculation procedure, and $f_{\rm f_{excr}} \geq 1$ is an empirical factor to allow for fuselage excrescence drag sources.

Specific values of interest are the momentum area $\Theta_{\rm wake}$ at the wake numerical endpoint $s_{\rm wake}$, the far-downstream momentum area $\Theta_{\scriptscriptstyle \infty}$, and the kinetic energy area $\Theta_{\scriptscriptstyle T\!E}$ at the body endpoint or trailing edge.

\[\begin{aligned} \Theta_{\rm wake}& = & \Theta (s_{\rm wake}) \\ H_{\rm avg} & = & {\textstyle \frac{1}{2}}\left[ H(s_{\rm wake}) + 1 + (\gamma\!-\!1) M_{{\scriptscriptstyle \infty}}^2 \right] \\ \Theta_{\scriptscriptstyle \infty}& = & \Theta_{\rm wake} \left( \frac{u_e (s_{\rm wake})}{V_{\!{\scriptscriptstyle \infty}}} \right)^{H_{\rm avg}} \\ \Theta^*_{\scriptscriptstyle T\!E}& = & \Theta^*(s_{\scriptscriptstyle T\!E}) \end{aligned}\]

The equation above is the Squire-Young formula, with $H_{\rm avg}$ being the average shape parameter between the end of the wake and far downstream.

The fuselage surface + wake dissipated power in the absence of BLI is then evaluated as follows, consistent with the usual wake momentum defect relations.

\[\begin{aligned} C'_{\!D_{\rm fuse}} & \equiv & \frac{\Phi_{\rm surf}-P_{V_{\rm surf}} + \Phi_{\rm wake}-P_{V_{\rm wake}}} {{\textstyle \frac{1}{2}}\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}^3 S} \hspace{6ex} \rm{(without BLI)} \hspace{-9.0ex} \\ C'_{\!D_{\rm fuse}} & = & \frac{D_{\rm fuse}}{{\textstyle \frac{1}{2}}\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}^2 S} \;=\; \frac{2 \Theta_{\scriptscriptstyle \infty}}{S} \hspace{17ex} \rm{(without BLI)} \hspace{-9.0ex} \end{aligned}\]

If BLI is present at or near the trailing edge, the upstream boundary layer and corresponding surface dissipation $\Phi_{\rm surf}$ will be mostly unaffected. But the viscous fluid flowing into the wake is now reduced by the ingestion fraction ${f_{\rm {\scriptscriptstyle BLI}_{\scriptstyle \,f}}}$, so that the wake dissipation $\Phi_{\rm wake}$ will be reduced by the same fraction. This then gives the following overall fuselage dissipation coefficient for the BLI case.

\[\begin{aligned} C_{\!D_{\rm fuse}} &\!=\! & \frac{\Phi_{\rm surf}\!-\!P_{V_{\rm surf}} \,+\, (\Phi_{\rm wake}\!-\!P_{V_{\rm wake}})(1\!-\!{f_{\rm {\scriptscriptstyle BLI}_{\scriptstyle \,f}}})} {{\textstyle \frac{1}{2}}\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}^3 S} \hspace{3ex} \mathrm{(with BLI)} \hspace{-2ex} \\ C_{\!D_{\rm fuse}} & \!\simeq\! & C_{\!D_{\rm fuse}} \,-\, C_{\Phi_{\rm wake}} {f_{\rm {\scriptscriptstyle BLI}_{\scriptstyle \,f}}} \hspace{23ex} \mathrm{(with BLI)} \hspace{-2ex} \\[0.5em] \rm{where} \hspace{3ex} C_{\Phi_{\rm wake}} & \!=\! & \frac{2 \Theta_{\scriptscriptstyle \infty}}{S} \:-\: \frac{\Theta^*_{\scriptscriptstyle T\!E}}{S} \end{aligned}\]

The induced drag is calculated using a discrete vortex Trefftz-Plane analysis. The circulation of the wing wake immediately behind the trailing edge is

\[\begin{aligned} \Gamma_{\!{\rm wing}} {\scriptstyle (\eta)} & = \frac{\tilde{p} {\scriptstyle (\eta)}}{\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}} \; \simeq \; \frac{p {\scriptstyle (\eta)}}{\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}} \, \sqrt{1 \!-\! \eta^{k_t}} \\ k_t & \simeq 16 \end{aligned}\]

where the approximation realistically represents the tip lift rolloff for typical taper ratios, and is consistent with the assumed $f_{L_{\scriptstyle t}}\simeq -0.05$ value for the tip lift loss factor. This circulation is convected into the wake along streamlines which will typically constrict behind the fuselage by continuity. The Figure above shows two possible aft fuselage taper shapes, giving two different wake constrictions.

An annular streamtube at the wing contracts to another annular streamtube in the wake with the same cross-sectional area. The $y$ and $y'$ locations on the wing and wake which are connected by a streamline are therefore related by the correspondence function.

\[\begin{aligned} y'{\scriptstyle (y)}& = & \sqrt{ y^2 - y_o^2 + {y'_o}^2} \end{aligned}\]

The Trefftz Plane circulation $\Gamma {\scriptstyle (y')}$ is then given by the coordinate shift. The mapping function $y'{\scriptstyle (y)}$ is not defined for $y \! < \! y_o$, so the circulation there is simply set from the $y_o$ value.

\[\begin{aligned} \Gamma_{\rm wake} {\scriptstyle (y')}& = & \left\{ \begin{array}{lcl} \Gamma_{\!{\rm wing}} \left( y {\scriptstyle (y')}\right) & , & y \!>\! y_o \\ \Gamma_{\!{\rm wing}} \left( y_o \right) \end{array} \right. \end{aligned}\]

The Trefftz Plane analysis uses point vortices. The circulation ([Gamwake]) is evaluated at the midpoints of $n$ intervals along the wake trace, spaced more or less evenly in the Glauert angle to give a cosine distribution in physical space. The wake's vertical $z$ positions are simply taken directly from the wing.

\[\begin{aligned} \theta_{i+1/2} & = & \frac{\pi}{2} \, \frac{i-1/2}{n} \hspace{2ex} , \hspace{2ex} i = 1 \ldots n \\ y_{i+1/2} & = & \frac{b}{2} \, \cos \theta_{i+1/2} \\ y'_{i+1/2} & = & \sqrt{y_{i+1/2}^2 - y_o^2 + {y'_o}^2} \\ z'_{i+1/2} & = & z_{i+1/2} \\ \Gamma_{\!i+1/2} & = & \Gamma_{\!{\rm wing}} (y_{i+1/2}) \end{aligned}\]

The locations of $n+1$ trailing vortices are computed similarly.

\[\begin{aligned} \theta_i & = & \frac{\pi}{2} \, \frac{i-1}{n} \hspace{2ex} , \hspace{2ex} i = 1 \ldots n\!+\!1 \\ y_i & = & \frac{b}{2} \, \cos \theta_i \\ y'_i & = & \sqrt{y_i^2 - y_o^2 + {y'_o}^2} \\ z'_i & = & z_i \end{aligned}\]

The circulations of these trailing vortices are the differences of the adjacent bound circulations, with the circulation beyond the tips effectively zero.

\[\begin{aligned} \bar{\Gamma}_{\!i} & = & \left\{ \begin{array}{lcl} \hspace{6.5ex} -\Gamma_{\!i-1/2} & , & i = 1 \hspace{3.5em} \mathrm{(left tip)} \\ \Gamma_{\!i+1/2}-\Gamma_{\!i-1/2} & , & i = 2 \ldots n \\ \Gamma_{\!i+1/2} & , & i = n\!+\!1 \hspace{2em} \mathrm{(right tip)} \end{array} \right. \end{aligned}\]

The above definitions are also applied to the horizontal tail, with its discrete points simply appended to the list and $n$ increased accordingly.

The Trefftz plane calculation proceeds by first calculating the $y$-$z$ wake velocity components at the $y'_{i+1/2},z'_{i+1/2}$ interval midpoints, induced by all the trailing vortices and their left-side images.

\[\begin{aligned} v_{i+1/2} & = & \sum_{j=1}^{n+1} \frac{\bar{\Gamma}_{\!j}}{2\pi} \left[ \frac{-(z'_{i+1/2}\!-\!z'_j)} {(y'_{i+1/2}\!-\!y'_j)^2 + (z'_{i+1/2}\!-\!z'_j)^2} \,-\, \frac{-(z'_{i+1/2}\!-\!z'_j)} {(y'_{i+1/2}\!+\!y'_j)^2 + (z'_{i+1/2}\!-\!z'_j)^2} \right] \hspace{2em} \\ w_{i+1/2} & = & \sum_{j=1}^{n+1} \frac{\bar{\Gamma}_{\!j}}{2\pi} \left[ \frac{ y'_{i+1/2}\!-\!y'_j} {(y'_{i+1/2}\!-\!y'_j)^2 + (z'_{i+1/2}\!-\!z'_j)^2} \,-\, \frac{ y'_{i+1/2}\!+\!y'_j} {(y'_{i+1/2}\!+\!y'_j)^2 + (z'_{i+1/2}\!-\!z'_j)^2} \right] \end{aligned}\]

The overall lift and induced drag are then computed using the Trefftz Plane vertical impulse and kinetic energy. The sums are doubled to account for the left side image.

\[\begin{aligned} C_{\!L_{\scriptscriptstyle T\!P}} & = & \frac{2}{{\textstyle \frac{1}{2}}\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}^2 S} \: \sum_{i=1}^{n} \: \rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}\, \Gamma_{\!i+1/2} \: (y'_{i+1}-y'_i) \\ C_{\!D_{\scriptscriptstyle T\!P}} & = & \frac{2}{{\textstyle \frac{1}{2}}\rho_{\scriptscriptstyle \infty}V_{\!{\scriptscriptstyle \infty}}^2 S} \: \sum_{i=1}^{n} -\frac{\rho_{\scriptscriptstyle \infty}}{2} \, \Gamma_{\!i+1/2} \left[ w_{i+1/2} \,(y'_{i+1}\!-\!y'_i) \:-\:v_{i+1/2} \,(z'_{i+1}\!-\!z'_i) \right] \end{aligned}\]

To minimize any modeling and numerical errors incurred in the wake contraction model and the point-vortex summations, the final induced drag value is scaled by the square of the surface-integral and Trefftz-Plane drag values.

\[\begin{aligned} C_{\!D_i} & = & C_{\!D_{\scriptscriptstyle T\!P}} \left(\frac{C_{\!L}}{C_{\!L_{\scriptscriptstyle T\!P}}} \right)^{\!2} \end{aligned}\]

This is equivalent to using the Trefftz Plane analysis to calculate the span efficiency rather than the actual induced drag coefficient.