Thrust and Power Required in Level Flight

Drag Polar Characteristic Points

Characteristic points 'E', 'P', and 'A' on the parabolic drag polar. The point 'S' is taken on the actual drag polar.

Thrust Technical Polar

Aircraft in wings level flight on a rectilinear, horizontal trajectory.

$\left\{ \begin{array}{l} T = D \\[2ex] L = W \end{array}\right. \label{eq:T:D:L:W}$ $L=W \quad \Longrightarrow \quad C_L = \frac{W}{\frac{1}{2}\rho V^2 S} \label{eq:Performance:CL:W}$ $T=D \quad \Longrightarrow \quad T \equiv T_\mathrm{R} \quad \Longrightarrow \quad C_D = \frac{T_\mathrm{R}}{\frac{1}{2}\rho V^2 S} \label{eq:Performance:CD:T}$ $\frac{T_\mathrm{R}}{W} = \frac{D}{L} = \frac{1}{C_L/C_D} = \frac{1}{E} \quad \Longrightarrow \quad T_\mathrm{R} = \frac{W}{E} \label{eq:Performance:T:Over:W}$

$T_\mathrm{R}$ 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.

Thrust technical polar. Example of thrust required for horizontal unaccelerated flight at a given altitude.

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

$V \propto \frac{1}{\sqrt{C_L}} \quad \Leftrightarrow \quad C_L \propto \frac{1}{V^2} \label{eq:Level:Flight:Speed:CL:Proportional}$

Airspeed in level flight and dependency on $C_L$ as well as angle of attack $\alpha$.

Drag in level flight as a function of true airspeed $V$. Angle of attack $\alpha$ necessary for vertical equilibrium.

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:

$V_\mathrm{s} = \frac{1}{\sqrt{\sigma}} \, V_{\mathrm{s,}\mspace{2mu}\mathrm{SL}} \label{eq:Stall:Speed:Altitude}$

Flight Mechanics for Pilots

Teaching material for student pilots of the Italian Air Force Academy

Thrust and Power Required in Level Flight

Drag Polar Characteristic Points

Thrust Technical Polar

Stall Speed

Drag