Fixed-Wing Stall Speed Equation Valid for Differing Planetary Conditions?

Question

For some time I have been looking for a simple stall speed equation that can factor in differing planetary conditions (gravity and air density) and changing vehicle mass, but have failed to find anything further than the standard stall speed equation:

v_stall = sqrt(2 * W/rho * S * C_L)

where,

v_stall = current stall speed

W = weight,

rho = air density,

S = wing area,

and C_L = max lift coefficient.

Additionally, I was looking for an equation that did not rely on wing area or lift coefficient. So, I derived my own expression. To be honest, I do not really remember how I came to it, I have been using it the past few years for hobby and conceptual design stuff, but it is the following:

v_stall = v_stall, 1g * sqrt( (1.225/rho_current) * (g_current/9.81) * (m_current/m_original) )

where,

v_stall = current stall speed

v_stall, 1g = stall speed under standard Earth conditions,

rho_ current = current air density,

g_current = current gravitational acceleration,

m_current = current vehicle mass,

m_original = original vehicle mass

It must be noted that v_stall, 1g is vehicle stall speed under 1 g and 1.225 kg/m^3 (standard Earth conditions)

This equation does work under theory and delivers results within a 1% to 3% difference from the standard equation. I ran calculations under many different air densities and gravitational accelerations, and the results almost always come startlingly close to the standard equation.

Can anyone else verify the validity of this equation?

Sorry for the poor formatting. I am new here.

Answer 1

Stall speeds vary widely based on many different factors, some of which you have listed already. However, an airfoil will always stall when it exceeds the critical angle of attack -- the point at which the CL or coefficient of lift reaches its maximum -- regardless of the configuration of the aircraft at any given moment. That is why pilots often fly to an AOA indicator first, and other factors second. Because of the way AOA works, air density is not relevant. Neither is gravity. The sensors (of various types) are used to calculate the angle of attack and tell you how close you are to stalling the wing. You may want to check out this link for a concise but very useful explanation of how AOA indicators work.

Answer 2

You are correct. The maximum coefficient of lift (CL_max) is non-dimensional, so it will be the same regardless of gravity, density, speed, mass etc. (it is not perfectly non-dimensionalized, as flow regime does still vary with Reynolds number, Mach number). Because CL_max doesn't change, we can say:

CL_max1 = CL_max2

Taking the stall equation:

CL_max = W / (0.5 * rho * v_stall^2 * S)

We can substitute it into the first equation (twice):

W1 / (0.5 * rho1 * v_stall1^2 * S1) = W2 / (0.5 * rho2 * v_stall2^2 * S2)

Rearranging and simplifying:

 v_stall2 = v_stall1 * sqrt(rho1/rho2 * S1/S2 * W2/W1)

If we assume that the area is unchanged, and we represent W2/W1 as mass*gravity, then this comes out to the same as your equation:

 v_stall2 = v_stall1 * sqrt(rho1/rho2 * g2/g1 * m2/m1)

These relationships hold true so long as we assume an identical Reynolds & Mach numbers. As you are looking at different planets, make sure your Knudsen number remains < 1e-4 too, otherwise, the atmosphere is too rarefied for this relationship to hold.

Answer 3

Yes, basically you just calculate a new stall speed for an airplane where mass, air density and gravity change, but wing area and maximum coefficient of lift remain constant. It is valid if similar airspeeds result in similar flow behavior (Reynolds number is the same), so the maximum coefficient of lift is actually the same. This may not be true if the atmosphere is made up of different gases, at a different temperature, or it is of such low density that you have to fly much faster and encounter compressibility effects. In this case the critical angle of attack also changes.