For calculating $V_{stall}$, why should be used $C_{L,max}$ and not $C_{L,min}$ (that would be more prudent)?

Question

Premise: In level unaccelerated flight we have relation $W=L=\frac{1}{2}\cdot\rho\cdot V_{stall}^2\cdot S\cdot C_{L,max}$ (taken from book Daniel P Raymer "Aircraft Design: A Conceptual Approach" equation (5.5) at page 85).

Question: Why $C_{L,max}$? Why not $C_{L,min}$? In fact in case the $C_L$ in use is not $C_L,max$, and speed is little more than $V_{stall}$ (valued for $C_{L,max}$), it happen that Lift is not enough and anyway stall happens on aircraft. If using $C_{L,min}$ the calculation is more cautious.

Answer 1

The stall speed is the speed at which you can still fly the plane in level flight. This is important, because strictly speaking, stall happens at a certain angle of attack, not at a certain speed. In other words, you can safely fly a plane way below stall speed - the problem is that you'll be doing it nose-down in order to keep the angle of attack to a minimum (and that as such you will quickly exceed the stall speed again - hey, we just did a stall recovery maneuver!)

So, let's see what happens when we approach stall speed from a higher speed, while flying level. As we slow down, we must increase our angle of attack to maintain level flight - this way, we are increasing the $C_L$ of the wing, as to keep the lift $L(=W)=\frac{1}{2}\rho VSC_L$ constant when reducing our $V$. Ideally, we'd like to keep increasing our $C_L$ as we reduce our $V$ - however, this means we keep increasing our AoA (angle of attack) until it gets past a critical point, where the airfoil stalls due to flow separation at the suction side of the airfoil.

So really, using $C_{L,min}$ would be nonsensical - if your wings would have a lift coefficient so low at such a low speed, I can guarantee you that $W>L$, i.e., you plummet from the skies. We need to increase our $C_L$ to maintain level flight, and the stall speed is exactly the point where we can no longer increase our $C_L$, or in other words, we arrived exactly at $C_{L,max}$.

Answer 2

There are multiple stall speeds that have been defined. For example, the 14 CFR §1.2 Abbreviations and symbols lists the following stall speeds:

$V_{S}$ means the stalling speed or the minimum steady flight speed at which the airplane is controllable.

$V_{SO}$ means the stalling speed or the minimum steady flight speed in the landing configuration.

$V_{S1}$ means the stalling speed or the minimum steady flight speed obtained in a specific configuration.

$V_{SR}$ means reference stall speed.

$V_{SRO}$ means reference stall speed in the landing configuration.

$V_{SR1}$ means reference stall speed in a specific configuration.

$V_{SW}$ means speed at which onset of natural or artificial stall warning occurs.

What you're asking is the $V_{S}$, the minimum flight speed at which the airplane is controllable. In short, as $V_{stall} \propto \sqrt{\frac{1}{C_{L}}}$, $C_{L_{max}}$ gives the minimum speed at which the aicraft is controllable.

Consider an aircraft in a level flight, If the pilot wants to reduce the speed, in order to maintain a steady, level flight, he/she has to increase the angle of attack i.e. increase the $C_{L}$. He can do this till the $C_{L}$ reaches $C_{L_{max}}$, where the speed becomes minimum while the aircraft is still in a steady, level flight. If the speed is reduced any further, the aircraft loses lift; this speed gives the stall speed of the aircraft.

You can get the stall speed for any configuration by using the $C_{L}$ at that configuration; but the values have to be realistic. Setting the $C_{L}$ to very low values to get large $V_{stall}$ makes no physical or practical sense. For example, the aircraft can be set to have zero (or even negative) $C_{L_{min}}$, in which case, the stall speed has no meaning.

Answer 3

In the equation $W=L=\frac{1}{2}\cdot\rho\cdot V_{stall}^2\cdot S\cdot C_{L,max}$ the goal is to find minimum flight speed which can produce lift equal to weight. This speed varies in different configuration but the minimum would be at maximum angle of attack producing $C_{L,max}$ (lift coefficient). The lift coefficient is a function of angle of attack, airfoil and wing geometry.

Answer 4

The general equation during level unaccelerated flight is $L=\frac{1}{2}\cdot\rho\cdot V^2\cdot S\cdot C_{L}$:

• For a variety of speeds V between the stall speed and the Never Exceed speed.
• For a variety of altitudes, between under sea level and the stratosphere.
• For a variety of Angle of Attack of the aeroplane, equating to a range of $C_L$
• Resulting in a variety of lift forces L.

There are three independent variables in the one equation, chose all three and the lift will be found, choose the lift and two others and the third one will be found.

This latter thing is what Raymer is doing. He sets lift to weight W for steady unaccelerated flight; he sets $C_L$ to $C_{L,max}$; and he then finds the minimum speed $V_{min}$ that the aeroplane can fly with at a certain altitude.

Note that for normal level unaccelerated flight $C_L$ is always > 0. The minimum $C_L$ is a negative value for negative AoA, and would mean that the aeroplane is flying upside down.