Wing Efficiency 

As previously mentioned, the goal with boost gliders and rocket gliders is to achieve the longest flight possible, which means achieving the longest glide possible. If the design approach not to simply boost a low efficiency glider as high as possible, it will be necessary to design a glider with as efficient a glide as possible. We think of this in terms of sink rate, defined as the rate at which a glider descends in still air in which not other forces, such as thermals are acting. Sink rate is measured in feet per second or meters per second.

In Appendix A, a mathematical method for determining sink rate and its derivation is presented. The formulae for sink rate is:

sink rate = sqrt (2*W / (S * r) ) * (Cd / Cl**3/2)

In this formula, W is the mass of the glider, S is the area of the wing, r (rho) is the air density, Cd is the drag coefficient and Cl is the lift coefficient. W/S is also the wing loading expressed in ounces per square inch or grams per square centimeter. Cl**3/2/Cd is a form of the wing efficiency factor, and is an indicator of the maximum lift that can be developed for a minimum of drag. It is important to the performance of any aircraft, including gliders, because the drag produced by the wing or induced drag is the vast majority of the drag found on the aircraft. The efficiency factor can be used to compare the efficiency of greatly differing types of wings. In the formulae for sink rate, the Cl is raised to the 1.5 power, thus resulting in a ratio of Cl**3/2 / Cd. This ratio can also be used as an efficiency factor; in fact, it is often referred to as the "Power Factor" [4], and will be referred to as such in this report. The basic efficient factor Cl/Cd will simply be called the Efficiency Factor.

So, we can use the sink rate formula to derive the wing efficiency or power factor of a given wing by knowing the wing loading of the glider and the sink rate it exhibits. As mentioned in the Introduction, we will use this wing efficiency factor to compare models with wings with different airfoils. If we test a number of gliders at the same time, the air density will be a constant, and so will make no difference.

Does the Reynolds number of a wing factor in to the efficiency of the wing? Absolutely. By looking at a couple of examples, we will find that the efficiency factor for large sailplanes is very different from that of a model sailplane.

The June/July 2000 issue of Quit Flight International, a British magazine of model sailplanes and gliders describes the old Schweizer SGS 2-32, a two person full sized sailplane. It has an area of 180 square feet and a flying weight of 1340 pounds or a wing loading of 7.44 pounds per square foot. Its minimum sink rate is 2.4 feet per second at 52 miles per hour. If we use a standard sea level air density of 0.0023769 lbs sec**2/ft**4, we can then calculate the wing power factor as described above to be about 33 for this

sailplane with a Reynolds number of more than 1 million. This for a sailplane that was considered an efficient trainer, rather than a high performance contest ship.

In the April/May issue of QFI, a high performance contest glider called Scar is described. It has a wing loading of 11.8 ounces per square foot, and a sink rate of approximately 1.5 feet per second. This results in a power factor of 16.6 for a model sailplane with a Reynolds number of about 150,000.

A typical A engine boost glider weighs about 1/2 ounces with an area of 40 square inches or a wing loading of 1.8 ounces per square foot. A sink rate of 2 feet per second would be considered good, resulting in a wing power factor or efficiency factor of about 4.86 for a glider having a Reynolds number of about 20,000. Even a sink rate of only 1 foot per second would result in an efficiency factor of less than 10.

We can assume that each glider of very different size is developed to produce the best glide possible. As the gliders get smaller and smaller, the wing loadings also get much smaller, and the sink rates should also get lower, but as we have seen, they do not because the wing efficiency or power factor goes down considerably. This is also true of exact scale model sailplanes. So, data from actual gliders again tells us that the Reynolds number has a big impact on wing efficiencies.

We can also see the fall off in airfoil efficiency in Figure 1. Figure 1 is a plot of Cl and Cd data taken for a typical airfoil. These types of plots with Cl versus Cd are used to design aircraft in various flight regimes.

In this plot, lift coefficient, Cl is on the left hand, vertical scale, while the drag coefficient, Cd is on the horizontal scale. The multiple curves are for the lift to drag plots at various Reynolds numbers. The left most and upper most curve is for a Reynolds number of 120,000, while the curve at the lower right is for a Reynolds number of 21,000. This plot is for an airfoil typically used in model aircraft, and was developed for the a wing that had a turbulator. A turbulator is usually (in models) a piece if string of narrow tape attached to the wing parallel to the leading edge, just behind the leading edge. It serves to cause the wing's boundary layer to become turbulent, thereby increasing the efficiency of the airfoil.

              Figure 1      A Cd - Cl Plot of a Typical Airfoil

In this plot we can see that the efficiency of the wing is much higher at the higher Reynolds numbers. For example, at a Reynolds number of 150,000, the minimum drag coefficient is .02 at a point where the lift coefficient is 0.6, resulting in an efficiency defined by Cl / Cd of 30. At a Reynolds number of 21,000 (typical of boost gliders), the minimum drag coefficient is .06 when the lift coefficient is 0.2, resulting in an efficiency factor of 3.33. Note that for the purposes of this R&D, I have computed the Power Factor, which is Cl**3/2 / Cd. At the minimum drag point for this airfoil in the curve for a Reynolds number of 21,000, the Power Factor would only be 1.5, and that would be just for the wing! If we take into account the drag for the whole glider, as we must for the simple experiments in this project, the Power factor would be even lower.

The maximum Efficiency Factor for a wing (only) with this airfoil would be about 6, and the Power Factor about 4.2. These maximum factors would occur in only a very narrow operational range for a glider using this airfoil. That is, the angle of attack would change just a little, and the efficiency of the wing would drop (along with the glider).

So, we see that typical airfoils used for large models and full size aircraft do not seem to have very high efficiencies at the Reynolds numbers typical of boost gliders and rocket gliders. Therefore, we cannot assume that the types of airfoils used in full size sailplanes or even model sailplanes are the best types to use with any size

Research Methodology The research methodology for this R&D project involves the use of three gliders about the size used for typical A engine boost glide competition. Each glider has a wing with a different airfoil. The gliders will be tossed by hand in straight and level flight, and timed from a measured height to determine the sink rate for each glider. Each glider will be tossed a number of times to eliminate any error caused by erratic tosses.

All expenses to perform this project are under $10.

All three gliders used for this project are very similar in design. All have similar wing spans and wing planforms. All have similar size horizontal and vertical tail surfaces and the same angle of incidence between the wing and horizontal tail surface. All have the same distance between the wing and horizontal surface, so that the moment arm should be the same. All have similar masses, and the slight differences can be taken into account using the formula for sink rate.

The specific data for the three gliders are as follows:

Table 1     Glider Specifications

Glider ID Wing span(inches)

Wing Thickness(inches)

Wing Area(inches**2)

Mass(ounces)

Glider Athick wing 14.0 0.25 34.04 0.485 (13.75 gm)

Glider Bcurved plate 14.8 0.25 39.80 0.377 (10.70 gm)

Glider Cflat plate 15.0 0.0625 40.45 0.383 (10.85 gm)

 All gliders have a root chord of 3 inches, and a distance from the trailing edge of the wing to the leading edge of the stabilizer of 4.5 inches. The incidence angle on each is 1.5 degrees. All are made of balsa, while the thick wing glider is built of balsa ribs and spars with tissue covering that has been lightly finished with clear dope. The think wing glider also has a spruce body rather than a balsa body. The thick wing glider has a horizontal stabilizer of about 8 square inches, while the other two have stabilizers of about 9.6 square inches. All have vertical stabilizers of about 2.5 square inches.

The major difference between the three gliders is the airfoil. As shown in Figure 2, the three airfoils differ greatly. The first airfoil, the "thick" airfoil is used for the wing of Glider A. It has a flat bottom, a rounded leading edge and a thickness of 0.25 inches or 8.33%. The second airfoil, the "curved plate" airfoil is made of 1/32 inch balsa curved to mimic the upper surface of the thick airfoil. The bottom surface is concave, because that surface is the bottom of the balsa sheet used to create the upper surface. The curvature is created using a small number of balsa ribs. The total thickness of the curved plate airfoil is 0.25 inches, and the leading and trailing edges have been sanded. The third airfoil is a flat piece of 1/16 inch balsa gently rounded at the leading and trailing edges.

Figure 2     Three experimental Airfoil Profiles

Figure 3 contains two pictures of the three gliders. From the left in both photos, the first glider is the thick airfoil glider with multicolored wing identified as Glider A. The second

glider is Glider B with the curved plate airfoil, and the third glider is Glider C with the flat plate airfoil.

The method for testing the gliders is to hand toss them in an open area and measure their flight duration from a measured height above the ground, thereby determining their sink rate. The ideal area for tests would be a large gymnasium or similar space, but none was available for these tests, and so the tests were performed outdoors. In order to decrease any effects from doing the tests outdoors, a large flat area in a park was selected, and tests were performed in the early morning hours when the ground is relatively cool and will not contribute any type of thermal activity. In addition, the tests were performed when the air was as still as possible.

Figure 3 Three Test Gliders

In the first step, the gliders are trimmed for straight and level flight and to achieve maximum flight times. After working with the gliders to insure reasonable trim, the throwing height is measured. Each glider is hand tossed a number of times to get consistent results. Each toss is timed for total flight duration. The gliders are tossed forward from shoulder height in a level throw that will immediately impart glide velocity to the glider without adding any altitude to the glider's position.

Because the best trim for each glider is unknown, even after initial trimming is performed for each glider, the nose weight of each glider is reduce by a small fraction of a gram a number of times during the tests. At no time was the nose weight of any of the gliders reduced to the point that a stall would result.

Each toss is documented with comments that indicate if the toss was inconsistent or out of level ("bad toss") or if any slight wind seemed to affect the glider's flight ("environmental factor"). The duration of the glide is noted in the log for each flight.

The tests were performed on a "cool" morning from 8 am to 9:30 am when the ambient air temperature was about 80 degrees F. The wind speed was virtually zero. The altitude of the park where these tests were performed is about 5700 feet.

The first tests were performed using a ladder that allowed a launch height of 84 inches. After a few tests of each glider were performed, it was determined that this height was insufficient to provide glides of sufficient duration such that the slight errors in timing could be absorbed. A location allowing a launch height of 134 inches was then used. Greater height would have been nice, but was unavailable.

Appendix C contains a table of all individual flights for the tests. for the purposes of calculating the efficiencies of the wings, the best three flights of each glider not affected by bad tosses or wind are being averaged in the summary table below. 

Table 2     Summary Of Flight Times

Glider ID Best Flt #1(seconds)

Best Flt #2(seconds)

Best Flt #3(seconds)

Average(seconds)

Sink Rate(ft/sec)

Glider Athick wing 4.0 3.7 3.7 3.8 2.94

Glider Bcurved plate 8.0 9.0 7.8 8.266 1.35

Glider Cflat plate 5.5 6.0 5.0 5.5 2.03

The sink rate equation can be slightly rearranged to identify the wing loading W/S:

sink rate = sqrt ( (W / S) * (2/ rho) ) * (Cd / Cl**3/2),

and using the average sink rate, determine the wing efficiency know as the power factor:

Cl**3/2 / Cd = sqrt ( (W / S) * (2/ rho) ) / average sink rate

Using a standard air density table [5], the air density at the test altitude is approximately 0.0019868 lb sec2 /ft4. The average sink rate and wing loading are used to calculate the Power Factors as shown in the next table. 

Table 3     Airfoil Efficiency

Glider ID Sink rate(ft/sec)

Wing Loading(oz/ft**2) Wing Power Factor

Glider Athick wing 2.94 2.05 3.86

Glider Bcurved plate 1.35 1.37 6.88

Glider Cflat plate 2.03 1.36 4.56

From this table, we see that the Power Factors calculated from the experimental data indicate the more conventional wing with the "thick" airfoil is the least efficient of the three. The flat plate airfoil is more efficient than the thick airfoil for gliders having a Reynolds number around 20,000. The curved plate airfoil is significantly more efficient than the thick airfoil. These results are discussed in the next section.

Conclusions 

This R&D effort illustrates that for the size and type of glider used in the experiments, a typical "thick" airfoiled wing with a flat bottom surface is probably not the most efficient. The flat plate airfoil operates just as efficiently, and is easier to build. This, in part, helps explain the good durations that simple boost gliders like the "Deltie" achieve.

This does not mean that the thick airfoil used in this R&D project is the most efficient that can be used. A thinner wing section may be more efficient or minute differences in the airfoil shape may improve efficiency considerably, but building such precise shapes in small wings becomes very difficult.

The curved flat plate wing used in Glider B is almost twice as efficient as the glider with the thick wing, as expressed by the Power Factor. The curved plate airfoil or a highly "undercambered" thin wing as described in Appendix B, seems to be the the most efficient for these small gliders. Gliders using this type of airfoil may be made even more efficient by changing the maximum thickness of the wing or by experimenting with the incidence angle between wing and horizontal tail surface.

There is a drawback to using these types of airfoils. The very thin sheeting used for Glider B's curved plate airfoil (1/32 inch balsa) will not stand up to the rigors of a rocket powered boost. I have tried a number of times with very inconsistent results. Other building techniques are available, but have yet to be used extensively, and may effect airfoil efficiency.

In addition, it should be noted that the curved plate wings may develop much higher drag than the other airfoils during the high speed boost. This could limit the boost altitude and therefore the total duration enough to negate any benefits of these airfoils.

Certainly, the results of this R&D project are encouraging, and warrant a "Phase 2" R&D project of some type. Future experiments will include wings of different thicknesses, gliders with different incidence angles, and curved plate airfoils of a construction that can stand up to rocket powered boosts. This may lead to boost gliders and rocket gliders that can attain significantly improved durations