Thrust and Power Required in Level Flight
Drag Polar Characteristic Points
Thrust Technical Polar

- $T_\mathrm{R}$ is the thrust required to fly at a given velocity in level, unaccelerated flight.
- Notice that minimum $T_\mathrm{R}$ is when airplane is flying at maximum $E$, i.e. at maximum $L/D$.
- $E=L/D$, aircraft aerodynamic efficiency is an important aero-performance quantity.
- $T_\mathrm{R}$ for airplane at given altitude varies with velocity.
Example of required thrust calculation, for given aircraft weight $W$ and flight altitude $h$:
- Select a flight speed $V$.
- Calculate $C_L = W / \big( \frac{1}{2}\rho V^2 S \big)$.
- Calculate $C_D$ from the polar: \(C_D = C_{D0} + C_L^2/\big( \pi \mathrm{A\!R} \, e \big)\).
- Calculate the efficiency $E = C_L/C_D$.
- Calculate $T_\mathrm{R} = W/E$.
$T_\mathrm{R}$ is is how much thrust engine must produce to fly at selected speed – at the given altitude and at the given aircraft weight. It coincides with the amount of drag at the selected flight conditions.
The speed in level flight at the value $C_L$ of the lift coefficient is expressed as follows:
\[V = \sqrt{\frac{2}{\rho}\strut} \, \sqrt{\frac{W}{S}\strut} \sqrt{\frac{1}{C_L}\strut} = \sqrt{\frac{2}{\rho_\mathrm{SL}\,\sigma(h)}\strut} \, \sqrt{\frac{W}{S}\strut} \sqrt{\frac{1}{C_L}\strut} \label{eq:Level:Flight:Speed}\]That is, in level flight
\[V \propto \frac{1}{\sqrt{C_L}} \quad \Leftrightarrow \quad C_L \propto \frac{1}{V^2} \label{eq:Level:Flight:Speed:CL:Proportional}\]Stall Speed
The minimum possible airspeed in level flight for given weight $W$ and flight altitude $h$:
\[V_\mathrm{s} = \sqrt{\frac{2}{\rho}\strut} \, \sqrt{\frac{W}{S}\strut} \sqrt{\frac{1}{C_{L\mspace{2mu}\mathrm{max}}}\strut} = \sqrt{\frac{2}{\rho_\mathrm{SL}\,\sigma(h)}\strut} \, \sqrt{\frac{W}{S}\strut} \sqrt{\frac{1}{C_{L\mspace{2mu}\mathrm{max}}}\strut} \label{eq:Stall:Speed}\]The stall speed varies with altitude as follows:
\[V_\mathrm{s} = \frac{1}{\sqrt{\sigma}} \, V_{\mathrm{s,}\mspace{2mu}\mathrm{SL}} \label{eq:Stall:Speed:Altitude}\]