Calculating Maximum Lift-Induced Drag
Aircraft experience two main types of drag: speed drag (or air resistance) and induced drag. The former is simply the drag that any object flying through the air will experience, that being the resistance that the air molecules provide to its motion. The latter is the drag induced by the lift of the aircraft. This happens because the high-pressure zone beneath the wing of the aircraft that provides its lifting force also provides a force backward on the plane, the magnitude of which depends on the speed of the aircraft, the density of the air it's traveling through, the wing profile, and the angle of attack at which the aircraft is flying.
The lift induced drag experienced by an aircraft can be calculated using this equation (detailed in this NASA article):
The induced drag coefficient can, in turn, be calculated using the following equation, detailed in this article:
These equations can be combined by substituting Cdi with the above expression, like so:
All of the values in this equation are constant except for air density, coefficient of lift, and airspeed, and since the induced drag force is directly proportional to all three of these values, this force will be greatest when all three are at their maximum value. Starting off easy, the maximum air density the aircraft will encounter is that at sea level, which has been recorded as 2.38 * 10-3 lb/ft^3. The velocity portion of this equation is also fairly simple, as, according to our magical spreadsheet of numbers, our aircraft will have a maximum operating velocity of 55 knots, which is approximately 92.83 ft/s.
The complexities of this equation begin to rear their ugly head when we examine our final value, the coefficient of lift. The website Airfoil Tools provided us with a way to estimate this coefficient at its absolute maximum. Given an airfoil configuration, determined by the Reynolds number that corresponds to the aircraft’s speed and chord width, it will generate a plot of Cl (lift coefficient) against the angle of attack, ɑ (see below). This is a numerical plot, and does not have an analytic representation, but it’s fairly trivial to find the maximum value of Cl on a given graph, as all of the data points are listed. The appropriate Reynolds number can be found using this equation:
Since our maximum airspeed of 55 knots is being used in the drag equation, it was used here as well. The chord width of the aircraft is 0.701 ft, or 0.214 m, and the kinematic viscosity of the air at cruise is 1.47 * 10-4 ft2/s, or 1.37 * 10-5 m2/s. This generated a Reynolds number of 442,676. The airfoil profile closest to this on Airfoil Tools used a Reynolds number of 500,000, which was close enough to make our calculations reasonably accurate. The this produced the following graph:
The maximum coefficient of lift on this graph was 1.4575, so this is the value that was used. Now having obtained all of the necessary values (again, many of these are in the magical spreadsheet of numbers), we can now plug and chug our drag equation. Here are the relevant values and the resultant equation:
It is here that I would like to note one of my biggest struggles in this calculation: the units. Solving the units in this equation yields units of ft-lbs^2, which confused me initially, to say the least, as drag force is a force, and so should have units of lbf (pounds-force). After some quick online research, however, it was pointed out to me that there was no problem at all, as the unit ft-lbs^2 is still a unit of force (just as how the metric unit of force, the Newton, is equivalent to kg-ms^2). However, this unit of force does not have a shorthand, as the shorthand unit for force in the imperial system, the lbf, is equal to the unit slug-fts^2, which is about 32.174049 ft-lbs^2. With this new knowledge (as well as even more reasons to hate the imperial system, as if I didn’t have enough already), I then proceeded to calculate the actual value of the drag force to be 11469.68 ft-lbs^2, which is equal to 356.49 lbf. This will allow us to move forward with the construction and testing of the wings and airframe.