A previous article 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 to measured value of Cd from the UIUC database, and then used to calculate drag, at each point across the span.
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 with airspeed. 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-1 illustrates the varying drag distribution with airspeed, that at different airspeeds there may be proportionately more or less drag near the tips. But bear mind, that while the drag increases with the square of airspeed, the drag coefficient tends to decrease with airspeed, until it reaches an optimum and then begins increasing again.
The above digression in drag variation with airspeed leads naturally into circling flight where the airspeed across the planform is not constant. This article will consider a wing at five bank angles: 0, 15, 30, 45, and 60. In this analysis, the comparison between the different bank angles keeps the average airspeed, and therefore the total lift, constant. This means that 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. 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-2 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. It shows that because of a decreasing Reynolds Number new the tips of the tapered planform, Cd increases. As the wing is banked to the left, the left wing tip travels through the air more slowly, and its Reynolds number decreases even further with a corresponding increase in Cd. The right wing tip, which travels faster in a bank, undergoes the opposite affect. Its Reynolds Number increases with a corresponding decrease in Cd at these airspeeds. See Figure-3 which shows the SD7037 airfoil Polar used in these examples.
At each increasing bank angle, the affect on Reynolds Number and Cd progresses. The Cd continues to increase on the left wing tip, as the bank angle increase, while the Cd decreases at the right wing tip. 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 45 o, and reverses once the bank angle is increased past 45 until it is again constant across the span at 90o. Therefore, beyond 45 o, the airspeed of the left wing tip starts increasing with a corresponding decrease in Cd.
Figure-4 similarly illustrates the total drag across the span. Even though Cd may be greater on the slower (inner) wing tip, the drag on a specific section of the wing is significantly less because it is related to the square of the velocity. However, as the surface area of the tapered wing decreases near the tip, the drag near the tip decreases relative to the inner sections of the outer wing half. 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. Induced drag is also greater on the outer wing tip in a bank.
The following table compares the total drag of a tapered and non-tapered planform at the various bank angles. 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. The table also shows that the drag increases with bank angle, increasing by roughly of 5.5%.
This and a previous article attempted to illustrate aerodynamic concepts quantitatively. This may help some better understand these concepts after learning them qualitatively. It important to recognize the which affects are greater, but the significance of these differences, because some are small. The previous articles emphasized the range of values where airspeed is constant across the wing span. This article showed the affects of circling flight where the airspeed varies across the span. It showed values at specific locations on the span and that even when the drag coefficient decreases, the drag along the span may increase due to greater spanwise airspeeds. For those interested in a tool for performing this type of analysis themselves, see http://ciurpita.tripod.com/rc/wing.