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.

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

where

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

and

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

Thus

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

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

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,

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

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

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

from which we find that

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

and

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

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

and

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