Aerodynamic and physical foundations of Power Required & Cruise.
In steady, unaccelerated level flight, an aircraft operates in complete force equilibrium: aerodynamic lift equals aircraft weight (L = W), and propulsive thrust equals aerodynamic drag (T = D). The total thrust required to maintain level flight is the sum of parasite drag and induced drag: T_R = D_0 + D_i = q * S * C_D0 + (W^2) / (q * S * pi * e * AR), where dynamic pressure q = 0.5 * rho * V^2. At low airspeeds, high lift coefficients generate severe induced drag; at high airspeeds, dynamic pressure drives parasite drag up quadratically. The minimum drag speed V_md occurs precisely where parasite drag equals induced drag (D_0 = D_i). The power required to sustain level flight is the product of thrust required and flight velocity: P_R = T_R * V = 0.5 * rho * S * C_D0 * V^3 + (2 * W^2) / (rho * S * pi * e * AR * V). Because parasite thrust scales linearly with dynamic pressure (T_parasite = q * f, where f = S * C_D0 is the equivalent parasite area), measuring thrust across calibrated dynamic pressure settings isolates the parasite drag area f. In this experiment, a scaled wind tunnel model of a business jet is tested across dynamic pressures from q = 0.60 to 4.00 kPa, plotting parasite thrust against dynamic pressure to determine equivalent parasite drag area f.
Viva voce questions & analytical theory
Why does the total thrust required curve T_R(V) exhibit a characteristic bathtub shape with a distinct minimum?
Total drag is the sum of induced drag (which decreases with velocity as 1/V^2) and parasite drag (which increases with velocity as V^2). At low speeds, induced drag dominates; at high speeds, parasite drag dominates. The sum creates a U-shaped curve with a minimum thrust required (minimum drag) at velocity V_md.
How do the minimum thrust speed V_md and minimum power speed V_mp compare?
The minimum power speed V_mp is always lower than the minimum drag speed V_md by a factor of 3^(-1/4) approx 0.76. At minimum power, parasite power equals one-third of induced power (P_0 = P_i / 3), making this the speed for maximum endurance in propeller aircraft.
What is the equivalent parasite area f of an aircraft and why is it useful?
Equivalent parasite area f = S * C_D0 is the cross-sectional area of an imaginary body having a drag coefficient of 1.0 that produces the same parasite drag as the entire aircraft. It provides an absolute measure of aerodynamic aerodynamic cleanliness independent of wing area.
How does increasing aircraft gross weight W affect the power required curve?
Increasing weight shifts the power required curve upward and to the right. Both the minimum drag speed V_md and minimum power speed V_mp increase proportionally to sqrt(W), and the power required increases proportionally to W^(3/2).
How does increasing flight altitude affect the power required curve?
Because air density rho decreases with altitude, the power required curve shifts to the right and upward in terms of True Airspeed. The minimum power required increases proportionally to 1/sqrt(sigma), but minimum drag remains constant in terms of Equivalent Airspeed.
What is the difference between maximum range speed and maximum endurance speed for a jet aircraft?
For a turbojet/turbofan where fuel flow is roughly proportional to thrust, maximum endurance occurs at minimum thrust required (V_md). Maximum range occurs where the ratio V / T_R is maximized, which is the point where a tangent line from the origin touches the T_R curve, corresponding to V_range = 1.316 * V_md.
Experimental precautions and wind tunnel safety
- Ensure sting support aerodynamic interference is calibrated and subtracted from raw axial balance drag.
- Verify that model pitch auto-trim accurately tracks L = W at every dynamic pressure step without hunting.
- Check that test section temperature remains stable to maintain constant air density and viscosity.
- Do not exceed the structural load limits of the 6-component internal balance during high-dynamic-pressure runs.
Theoretical assumptions & flow limits
- Assumes parabolic drag polar without Mach compressibility drag rise (valid for M < 0.65).
- Wind tunnel Reynolds number is lower than full-scale flight, requiring standard flat-plate skin friction scaling corrections.
- Engine nacelle internal flow through-ducts create internal drag that must be accounted for via momentum tare.
- Assumes zero flight path climb angle (steady level flight gamma = 0).