Burt Rutan's White Knight and SpaceShip One, Photo Courtesy of Scaled Composites
Thermodynamics and Propulsion

13.1 Vehicle Drag

Recall from fluids that drag takes the form shown in Figure 13.2, being composed of a part termed parasitic drag that increases with the square of the flight velocity, and a part called induced drag, or drag due to lift, that decreases in proportion to the inverse of the flight velocity.

Figure 13.2: Components of vehicle drag.
Image fig4DragComponents_web

$\displaystyle C_D = C_{D_0} + \frac{C_L^2}{\pi e AR},$    


$\displaystyle L = \frac{1}{2}\rho V^2 S C_L$    


$\displaystyle D = \frac{1}{2}\rho V^2 S C_D.$    


$\displaystyle D$ $\displaystyle = \frac{1}{2}\rho V^2 S C_{D_0} + \cfrac{L^2}{\cfrac{1}{2}\rho V^2 S}\left(\frac{1}{\pi e AR}\right)$    


$\displaystyle D$ $\displaystyle = \frac{1}{2}\rho V^2 S C_{D_0} + \cfrac{W^2}{\cfrac{1}{2}\rho V^2 S}\left(\frac{1}{\pi e AR}\right)$    

The minimum drag is a condition of interest. We can see that for a given weight, it occurs at the condition of maximum lift-to-drag ratio,

$\displaystyle D = L\frac{D}{L}= W\left(\frac{D}{L}\right)=W\left(\frac{C_D}{C_L}\right).$    

We can find a relationship for the maximum lift-to-drag ratio by setting

$\displaystyle \frac{d}{dC_L}\left(\cfrac{C_{D_0} + \cfrac{C_L^2}{\pi e AR}}{C_L}\right) = 0$    

from which we find that

$\displaystyle C_{L,\textrm{min drag}} = \sqrt{\pi e AR C_{D_0}}$    


$\displaystyle C_{D_\textrm{min drag}} = 2 C_{D_0}$    

$\displaystyle \left(\frac{C_L}{C_D}\right)_{\textrm{max}} = \frac{1}{2}\sqrt{\frac{\pi e AR}{C_{D_0}}}$    


$\displaystyle V_{\textrm{min}} = \sqrt{\cfrac{W}{\cfrac{1}{2}\rho S C_{L,\textr...
...1}{\rho^2}\frac{1}{C_{D_0}}\left(\frac{1}{\pi e AR}\right)\right]^{\frac{1}{4}}$