# Why Lift-induced Thrust Required(Tr) decreases and Zero-Lift Tr increases with the increase of velocity? [duplicate]

As the minimum Tr is at the point where L/D (lift/drag) ratio is the most. And Lift-induced Tr depends on the Coefficient of Lift, (Cl)^2 and Dynamic Pressure (q) which also increases as velocity increases. But why it is decreasing?

And how Zero-Lift Tr depends on velocity and is affected that much when its coefficient is a constant?

• I am offering aviation.stackexchange.com/questions/36062 as duplicate. It asks generally why the induced drag formula looks like it does, and contains a detailed derivation. Apr 20, 2021 at 19:01
• There are also How is the induced drag calculated for a wing with elliptical planform? and Is the induced drag independent of wing span? with further derivations of induced drag. Apr 20, 2021 at 19:05
• @JanHudec, thank you but I had this formula already. I am grateful that you reminded me it was a level, unaccelerated flight. What I needed is a formula for zero-lift Tr --while its coefficient is a constant, its formula is not: it actually has a cube of velocity. as said by @zaitcev Apr 20, 2021 at 19:06
• Apr 21, 2021 at 8:44

This happens because Angle of Attack required to maintain flight at a given weight decreases with the increase of speed.

As for the zero-lift Tr, also known as "parasitic drag", while its coefficient is a constant, its formula is not: it actually has a cube of velocity.

• Yes, I am aware of the fact that AOA decreases, but how it relates to increasing Zero-lift Tr? I am sorry if it is very simple yet I am not understanding. Edit: Found your reason later, may you please direct me to the formula. Thanks. Apr 20, 2021 at 17:01
• re "it actually has a cube of velocity."-- can you expand on this? Apr 20, 2021 at 17:48
• There is no cube of velocity in any drag. The aerodynamic formulae (except when dealing with compressibility) only include velocity in dynamic pressure terms $q = \frac12\rho v^2$, so velocity can only appear in them in even powers. Apr 20, 2021 at 19:43

Thanks to @JanHudec for reminding me that the graph is for a level flight.

To keep the flight level at low speeds, the Angle of Attack(AoA) needs to be increased, meaning Cl (coefficient of lift) must be increased and so the lift-induced drag ( proportional to Cl^2) is high at low speeds.

With the increase of speed, AoA decreases to keep the flight level, hence, Cl and so lift-induced drag and its Thrust Required (Tr) decreases.

Coming to Zero-lift Drag or the parasitic drag. At low speeds it is low. But at higher speeds, there is an introduction of wave drags,

"parasitic drag increases because the fluid is striking the object with greater force, and is moving across the object's surfaces at higher speed. As speed continues to increase into the transonic and supersonic regimes, wave drag grows in importance."-Wikipedia.

Therefore, in the case of Zero-lift, the increase of velocity is not much of the reason (by this I mean, not the proportionality with the square of velocity), but the increase of wave drag is. And so there is not much effect on Induced drag with the increase of velocity