An article titled Numerical Understanding of Planforms in the November 2003 of RCSD showed how the Vortex Lattice Method and the UIUC Airfoils database were used to calculate the lift and drag at each point on the wingspan. Using the power of the PC, it focused on the actual values of lift, Cl, Cd, and drag at each point across the wing, and the total drag, rather than average values. It showed how the calculated values of lift and Cl, can be related to measured values of Cd from the UIUC database, then used to calculate drag, at each point across the span, and finally the total profile drag. It does not discuss induced drag which is proportional to lift.
The previous article focused on the steps involved in the analysis, not a valid comparison of planforms. It very crudely compared a tapered and non-tapered planforms, showing little (<2%) difference in total drag between the two. However, this comparison may be misleading since it was only at one airspeed.
The following table shows values for each planform at several other airspeeds. These additional airspeeds do not produce lift equal to the weight of the aircraft. It shows that at two airspeeds, the non-tapered planform actually has less drag than the tapered planform, and one was used in the comparison in the previous article.
The reason for the varying results in the table is the non-linear behavior of drag at low Reynolds Numbers. See Figure-1. In other words, a slight change a airspeed may lead to significant changes in drag that are not proportional to the change in airspeed. This same behavior applies to a tapered planform with constant airspeed across the span.
Figure-2 illustrates the varying drag distribution with airspeed. Notice how the drag changes erratically near the wing tips where the Reynolds Number is smaller. But bear mind, that while the drag increases with the square of airspeed, the drag coefficient tends to decrease as the airspeed and Reynolds Number increases. In the results shown in Figure-1, the wing is divided into 32 panels, each panel representing a section of the wing from leading to trailing edge. The indicated drag is the total drag corresponding to a panel. The total drag produced by the wing is the sum for all panels.
The above digression in drag variation with airspeed leads naturally into circling flight where the airspeed across the planform is not constant. This article considers a wing at five bank angles: 0, 15, 30, 45, and 60 degrees. In this analysis, the comparison between the different bank angles keeps the average airspeed constant. If the airspeed for the aircraft with zero bank angle generates sufficient lift to support the weight of the aircraft, then at any bank angle greater than zero, there will be insufficient lift to support that weight. This, therefore, compares non-realistic flight conditions, and is only useful in this comparison of drag.
In this analysis, the aircraft is banked to the left, the left (inner) wing tip is lower than the right, and the aircraft is circling counter-clockwise. The airspeed of the inner wing-tip is therefore slower and the outer wing tip is faster than the average airspeed.
Figure-3 illustrates the drag coefficient (Cd) across the span for the tapered planform at each bank angle. The zero bank angle is symetrical and serves as a reference. As the wing is banked to the left, the left wing tip travels more slowly and its Reynolds number decreases. The drag polar for the SD7037 airfoil shown in Figure-1 shows a signficant Cd penalty at the lowest Reynolds Number. The right wing tip, which travels faster, undergoes the opposite affect. Its Reynolds Number increases with a corresponding decrease in Cd.
While five bank angles are considered, only four appear in the plot because the data is the same for bank angles of 30 and 60. The airspeed difference at the wing tips is a maximum at 45o, and reverses once the bank angle is increased past 45 until it is again constant across the span at 90o.
Figure-4 It shows that even though the drag coefficient increases at the left wing tip as the bank angle increases and its airspeed slows, the actual drag decreases because it is related to the square of the airspeed. Just as in non-circling flight the actual drag may decrease near both tips as the chord and wing area decrease. So while the aircraft is circling counter-clockwise there is more drag on the outer wing tip, even though it may be more efficient in terms of Cd.
Table-2 compares the total drag at various bank angles, and between the tapered and non-tapered planforms. The tapered planform has less drag than the non-tapered planform for all bank angles greater than zero that are shown. It also shows the symmetry around the 45 o bank angle, where the total drag is the same for bank angles of 30 and 60. And it also shows that the drag increases with bank angle, increasing by roughly 5.5% at 60o.
However, while the above table serves as a reference of how drag is affected when the plan is circling, it does not model realistic flight conditions While maintaining a constant airspeed maintains the total lift, this lift is no longer vertical as the bank angle increases. Greater lift is required as the bank angle is increased to support the weight of the aircraft, and most likely will result in even greater drag.
Further, more realistic comparisons are complicated by the number of mechanisms involved to establish stable circling flight, including changes in airspeed, angle-of-attack and trailing-edge deflection (flap and/or aileron). And even though the airspeed is different on each wing, both wings must produce equal lift to prevent rolling. Aileron, dihedral and aircraft yaw asymetrically affecting the angle-of-attack on each wing, can all be used to balance the lift on each wing.
This and the November article hopefully illustrate aerodynamic concepts through numerical comparisons. This may help some better understand these concepts. For the examples used, it showed which concepts had the greater affect, but also, in many cases, how little the difference is. For those interested in a tool for performing this type of analysis themselves, see http://ciurpita.tripod.com/rc/wing. I'd like to thank Dave Register for the help provided in this and previous articles.