Matching the Drag Equation to Real Airplane Data

Question

I am trying to compute airplane performance for a piston-driven airplane with constant speed prop using the standard equations for Lift, Thrust and Drag. I have found that there are a couple of useful variables that I can change to make the results more closely match with published data. These are Cd0 (the coefficient of Parasitic Drag) and K, a multiplier used to compute Cdi (the coefficient of Induced Drag). Cd0 and Cdi are added to together to compute Cd (the total coefficient of Total Drag), which is used to compute the Total Drag Force.

Most of the value of K is given by the formula: K = 1/(Aspect Ratio (AR) * Wing Efficiency (e) * PI). But you can further adjust K by multiplying it by a another constant, which I shall call K0.

You can enter estimated values for Cd0 and K0 and see if the results agree with the published performance data for the aircraft. I generally try to use Sea Level data since that eliminates the variances that arise at different altitudes. I also need to take care to determine the aircraft Weight or Mass that is used in computing the public data.

DATA POINTS

At Maximum Speed (Vmax), Drag should equal Maximum Thrust (Tmax). On a piston or propeller aircraft Tmax decreases with Speed, so you have to use the trial and error method to compute Vmax.

At Maximum Lift/Drag Speed (L/D max), Cdi = Cd0. This is the low point for Cd and Total Drag. There are equations for computing the Speed at L/D max.

At the Best Rate of Climb Speed (Vy) for a piston-driven airplane with a constant speed prop, Cdi = 3 * Cd0. There are equations for computing the Speed at Vy speed.

My understanding is that you must use the trial and error method to determine the Best Angle of Climb Speed (Vx) for a piston-driven airplane with a constant speed prop and I am not aware of any necessary relationship between Cdi and Cd0.

At Takeoff Speed, the coefficient of Lift (Cl) = Clmax. I am not aware of any necessary relationship between Cdi and Cd0.

METHODOLOGY

I create an Excel spreadsheet which has a graph like this:

I initially leave K0 at 1 and insert different values for Cd0 until I find one where the lines for Tmax and Total Drag cross at the published Vmax.

I determine whether the lines for Parasitic Drag and Induced Drag cross at the published speed for C/L max. If not, I adjust Cd0 and K0 until they do, while not changing the computed Maximum Speed.

I determine whether, at published Vy, Cdi = 3 * Cd0. If not, I adjust Cd0 and K0 to try to create the best fit.

I can change the Thrust and Drag values a bit by changing Prop Efficiency and Wing Efficiency, respectively.

Given that the Drag Equation follows a fairly fixed plot, there may be situations where I get two of the data points to fit perfectly, but a third data point does not - which means that I must use values which generate the "best fit".

The computations in the chart are for an FM-2 Wildcat, which has a top speed of 300 mph and a Vy of 144 mph. I used value of .01825 for Cd0 and 2.74 for K0. The results match the published data almost perfectly (but I only found 2 data points so that is inevitable).

I am not taking including propeller efficiency (Np) into account. I expect that the propeller efficiency should reach the max value before takeoff speed and will remain at that max value until just before Vmax. So, it might reduce the actual Vmax from what I have computed.

ADDENDUM

From another I learned that "The ratio between max range and max endurance speeds is always 3^0.25." Since the Max Range Speed equals the L/D max Speed, you compute Max Endurance Speed by dividing L/D max Speed by 3^.25. It turns out that Max Endurance Speed equals the Vy Speed. So this provides a mathematical connection between the two.

Also, reviewing the effect of changes in altitude, it appears that for both Vy and L/D max, the Cl and the relationship between Cdi and Cd0 remain unchanged as altitude increases. And Vy Speed remains equal to L/D max Speed divided by 3^.25.

QUESTIONS

1. While not perfect, does this methodology make sense?

2. Are there other data points I can use?

3. Are there any other speeds, besides L/D max and Vy where there is a fixed relationship between Cdi and Cd0?

Answer 1

What you describe is commonly called the quadratic drag equation.

1. While not perfect, does this methodology make sense?

Yes, see answers here, here and here, for example. But you can easily refine the results. It helps to adjust zero-lift drag with speed, for example, as done in this answer. This should give better results than your approach with K0. Your zero-lift drag coefficient is almost certainly too low.

2. Are there other data points I can use?

Yes, anywhere you can determine drag precisely. This is harder than it sounds, however. Normally you will only get a combination of drag and thrust. Once thrust is known exactly, the drag coefficient can be calculated if you know the airplane weight, load factor, flight speed, rate of acceleration, climb speed and the atmospheric conditions for this data point.

3. Are there any other speeds, besides L/D max and Vy where there is a fixed relationship between Cdi and Cd0?

No, not the way you desire.

I do not know how you calculate thrust, but I am sure you underestimate the drag increase due to prop wash. Here is an answer with a link to a very interesting article on the subject. Also I wonder whether you included the effect of constant CAS on climb speed. Close to stall speed the quadratic drag equation loses some of its shine due to separated flow beginning to nonlinearly increase drag. Here it is best to assume a linear lift curve slope when calculating the induced drag coefficient but to gradually reduce the lift coefficient you use for speed calculations as stall speed is approached.