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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

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

where

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

and

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

Thus

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

or


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

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

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

and

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

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

and

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

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