# What is the constant in this formula for additional lift?

Let’s consider the case of a plane flying s/l in a headwind with a positive gradient. A wing produces lift by inducing a bound vortex superimposed on the relative wind., and in the particular case of flying in a positive gradient, one can imagine an extra pseudo-vortex associated with the gradient itself, that re-inforces the bound vortex producing additional lift.

The question is now how to quantify that additional lift. From the variables that enter in the circulation theory of lift, one can guess that the variables involved are the density of the air rho, the airspeed v, the wing area A, the chord c and of course the wind gradient ß.

From dimensional analysis, it results that the extra lift L is:

L = k · rho · v · A · c · ß

where k is a constant to be determined. It can be probably done with a ‘material experiment’, but my question here is whether it can be derived from already known data or from a ‘mental experiment’.

• See grc.nasa.gov/www/k-12/airplane/lifteq.html and grc.nasa.gov/www/k-12/airplane/liftco.html, describing some of the issues. – Řídící Apr 16 '17 at 14:08
• @xxavier, do you mean the angle of attack when you actually say gradient? – GHB Apr 17 '17 at 9:20
• @GHB No, of course I don't mean that... The gradient of the wind is the variation of wind speed with altitude. For example, if you have 20 m/s variation in an altitude difference of 100m, the gradient is 20/100 m/s/m = 0,2 s^-1 – xxavier Apr 17 '17 at 11:34
• Why do you think that your formula is correct? There are a lot of unexplained assumptions in it. With the lift and drag equations of the first comment's link you should be able to get a formula using a physical model instead of guessing. – Gypaets Apr 18 '17 at 9:08
• @Gypaets The variables are the relevant ones in the problem, in particular rho, v, A, and ß; with c added because of dimensional considerations. I have already found two possible solutions, one of them presumably exact (π/2) and the other is an approximate solution that differs by only 5%. But I'm not sure... Hence my question... – xxavier Apr 18 '17 at 10:28

• Lift is not determined by the sum of wind and aircraft velocities (measured relatively to the ground). Only relative velocity of air (air velocity relatively to the aircraft). So there is no alpha change according to wind velocity relatively to the ground (what you name $V_w$), and neither change according to aircraft velocity relatively to the ground ($V_k$). Or maybe I don't understand your (untold) definitions of these velocities. – mins May 23 '17 at 18:58