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The Flight of a Balsa Glider

Chris Waltham

Department of Physics and Astronomy, University of British Columbia, Vancouver BC, Canada V6T 1Z1 (waltham@physics.ubc.ca)

Published in the American Journal of Physics 67 (1999) 620-624

Abstract A simple analysis is performed on the flight of a small balsa toy glider. All the basic features of flight have to be included in the calculation. Key differences between the flight of small objects like the glider, and full-sized aircraft, are examined. Good agreement with experimental data is obtained when only one parameter is allowed to vary. Introduction Small gliders made of paper or balsa wood are frequently used in physics classes to demonstrate the balance of forces and features of fluid mechanics such as Bernoulli's Theorem. One immediate observation is that all gliders - balsa or paper - have about the same glide slope, about 1 in 4 , which coincides neatly with the angle of the seating in a typical lecture theatre. This seemingly innocuous feature is however at odds with the behaviour of larger models and full-sized aircraft, the shape of whose wings critically affects the glide slope. The natural flying velocities of small gliders are also very much alike, a few m/s. Presented here is a quantitative analysis of the flight of a small glider, a “Guillow Super-Ace” ($1.99) made of sheet balsa, of a sort which can be bought in any toy or model store (see specifications in table I). Out of the analysis will appear broad features which are observed in the flight of all hand-held gliders. Flight is achieved by the interaction of a vehicle with the air surrounding it. As the aircraft moves through the air the flying surfaces deflect air downwards creating a force which can be resolved into components perpendicular to its motion (lift), and parallel to its motion ("induced" drag). At the same time the flow of air past the wings and body of the craft is slowed by friction and changes in pressure caused by the shape; this causes more resistance called friction and pressure (or form) drag respectively. A well-trimmed glider flies in a straight line at a constant speed, necessarily in a slightly downward direction, by balancing the forces of lift and drag with that of its weight. The flight path is determined by the relative sizes of the two types of drag force (fig 1.). Aerodynamics at Low Speeds The design criteria for a balsa glider are as follows. The flight speed should be fast enough so it will not be unreasonably affected by normal indoor turbulence (< 1m/s) but not so high as to be dangerous (>10 m/s). These constraints affects the mass, which as we

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will see below, is in the few g range for small hand-held aircraft, making balsa wood or paper appropriate materials for construction. Any discussion of size and speed regimes in aerodynamics leads us immediately to ask about the Reynolds number Re as this will tell us what type of flow we are dealing with.

Re = ρ Vx / µ (1) In the above equation ρ is the density of air, V the flow velocity, µ the viscosity and x the relevant length scale, frequently the chord of the wing. The values of ρ , V, and µ refer to values a large distance from the moving object. In the case of air at STP, Re becomes in SI units:

Re = 68,000 Vx (2) The Guillow Super Ace flies best at ~ 4 m/s and it has a mean chord c = 37 mm, so Re (for the wing) is ~10,000. The Reynolds number hierarchy of flying objects is as follows: • Insects: 102-104

• Model Aircraft: 103-106 • Birds: 104-106 • Aircraft: Human and solar-powered 105-106 ; Light 106-107 ; Large 107-108. Aerodynamically this places our glider at the lower end of the model and bird regime and close to that of insects. It takes us into the laminar flow regime, where drag is high and lift poor1, which occurs below Re ~ 300,000. The lift force L is understood in terms of speed, air density, plan wing area (S) and a coefficient of lift CL . The expression for lift has the following form; for flight at constant velocity at angle θ to the horizontal the lift of the wing must equal the component of the weight of the aircraft W perpendicular to the flight path (fig.1).

L = ½ CL ρ V 2 S = W cosθ (3) For a wing with a thin airfoil section, aspect ratio (span / mean chord) A , at incident angle α, with an attached boundary layer2:

CAL = ⋅

+λ

π α( )

sin/

Re21 2

(4)

The value of λ (Re) approaches unity for Re > 106 , but for our flight regime (Re ≈ 104), it rather less. Equation 4 is also true for a flat plate, but for a much lower smaller range of angles than for more appropriately shaped airfoils. Above α ~ 10°, CL starts to drop below the ideal value, and above ~ 14°, it drops precipitously (i.e. the wing stalls). Hard data is rare; flat plate wings have not been very interesting since Horatio Phillips, in the

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last years of the 19th century, found cambered ones to be better. Bradley Jones’ textbook3 of 1942 shows a useful plot for a flat plate with A = 6 (very close to the glider’s value of 5.25) at about the right Re (fig.2). This can be parameterized with a fair degree of accuracy using an exponential to describe the stalling behavior:

CL = − °4 0 0 0037 35. sin . exp( / . )α α (5) Similarly for the drag force D:

D = ½ CD ρ V 2 S (6) The drag coefficient CD is made of up of three terms, friction, pressure and induced. The first, for flat sheets parallel to the fluid flow, in the laminar region, is given by the expression first derived by in 1908 by Blasius4 :

CD f,.. ~= <−2 66 500 0000 5Re Refor , (7)

The second term, pressure drag (coefficient CD,P ), is zero for ideal flat sheets. We handle any deviation from ideality by multiplying the friction drag coefficient by a factor f . The third term, induced drag, can be handled simply for flat sheets, because the total force on the sheet is a result of a pressure difference above and below the sheet, and so it has to be normal to the surface5 (fig. 2). The diagram shows the net force as passing through the center of the sheet. In fact the force intersects the sheet somewhat forward of this point as can be observed by trying to “fly” a wing made of a piece of card; it spins about its long axis with the leading edge rising. However, this position shift has no effect on the direction of the induced drag. The induced drag can be inferred from the lift, and has the same dependency as lift on ρ ,V and S .

Di =L tan α ; CD,i = CL tan α (8) Thus for small α , CD,i is approximately proportional to CL

2, and therefore only slightly dependent on the aspect ratio A . For good airfoils at high Re, the expression for induced drag is as follows:

CC

AD iL

, ≈2

π (9)

Thus the dependence on A is much stronger, and this favors long, thin wings for commercial aircraft and full-sized gliders. In all cases, the total drag is given by:

D C V S C C C VD D f D P D i= = + +12

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2 2ρ ρ( ), , , S (10)

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We can use these equations to optimize whatever flight quality we wish, in particular range and therefore glide angle. Range is how far an aircraft can fly with a given amount of energy (chemical energy in the fuel for a powered aircraft, potential energy for a glider). The glide angle (see fig. 1) is θ , the angle the flight direction makes to the horizontal, and it is related to the ratio of lift to drag. This ratio is at a maximum, (L/D)max, when the angle is at a minimum, θ0 . For small angles of attack (tan α ≈ sin α) we can estimate (L/D)max using equation 7:

L D C CA

A CL DD f

/ /sin / ( / )

sin / ( / ) ,

= ≈⋅ +

⋅ + +λ π α

λ π α2 1 2

2 1 22 (11)

This can be maximized by differentiating with respect to sin α . The maximum occurs when the two terms in the denominator are equal, i.e. friction drag equals induced drag.

( / ) ( / )/ ( / )

max max,

L D C CA

CL DD f

= ≈⋅ +λ π 1 2

2 (12)

Note the very weak dependence of (L/D)max on aspect ratio, velocity and chord, from which we can expect a fairly universal value for all hand-held gliders. We can see what this is from an estimate of the performance of our particular glider, which from experiment flies best at Re ≈ 10,000. The Blasius formula gives the drag coefficient, which is equation 7 multiplied by the total surface area divided by the wing area (a factor 1.70 in this case). The aspect ratio A is 5.25 and from the parameterization of Jones’ data (eqn.5) λ ≈ 0.9. This yields (L/D)max ≈ 5. Using this result and equations 3,4 and 6 gives v0 ≈ 5 m/s. This velocity depends on the following parameters:

vW c

S03 2 0

1 2/

/sin∝

θ (13)

Hence we can see for a hand-held glider of a few grams, the natural flying speed will not deviate from the range of several m/s. This simple analysis comes close to our measured values of (L/D)max = 4.25 at v0 = 3.7 m/s, and the discrepancy indicates that the friction drag is greater than for ideal flat sheets, and/or that the pressure drag is non-zero. In “normal” aerodynamics, i.e. good airfoils at high Re, the expression for (L/D)max is as follows. Note again the much stronger dependence on aspect ratio.

( / ) ( / )/ ( / )

max max, ,

L D C CAC CL D

D f D P

= ≈++

12

1 2π A (14)

Data and Analysis

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The experiment was performed in a large, steep, empty lecture theatre. The glider was thrown many times and great effort was made to launch it at about the right angle and speed to give it a linear trajectory. The flight was timed and the impact point was noted so that v and θ could be obtained using a measuring tape and a theodolite. The flight path was 25 m long. The position of the wing in the fuselage was varied to optimize the glide slope. The process was not easy and only a small fraction of the flights were straight enough to give good data; this was particularly true below 3.3 m/s and above 5.5 m/s. The results are shown in fig. 4. Modeling the flight of the glider required a straightforward iterative calculation. There was only one free parameter, the drag factor f , which was expected to be close to unity. The steps were as follows: • Start with angle of attack α ; allow it to vary from 0 to 10° in 0.1° steps. • Calculate the lift at required at this angle, Wcosθ , (start with θ = 90° for α = 0) • Guess a starting value for the total drag coefficient, CD . • Determine Re values for the wing, fuselage, stabilizer and fin. • Calculate the drag for each member as ( )f × −2 66 0 5. .Re . • Find the induced drag. • Add all the drags to find the total. • Find (L/D) for each value of α , and hence find θ . Feed to next value of α . When the calculation was done for all values of α : • Read off (L/D)max and find v0 . • Vary f to reproduce the experimental values. • Check the results are independent of the choice of initial CD . One implicit simplification should be noted: we have assumed, without mathematical justification, that there is no interference between the components of the glider, which have been treated separately. The wings will have a small effect on the drag of the fuselage, as they will on the airflow around the tail. The result for the drag factor (de-emphasizing the data points at extreme velocities) is

. The value is close to unity. There will be an increase in drag caused by blunt leading edges and rough surfaces. There will also be a decrease in drag due to the separated flow over the top of the wing suffering less friction than the Blasius formula (eqn. 7) would indicate

f = ±16 0 2. .

5. Determining which effect is greater is beyond simple theory. Our analysis with f = 1.6 gives (L/D)max = 4.06 at v0 = 4.05 m/s (Re = 10,000) and α = 6.5°, with a total drag coefficient, CD = 0.12. The contributions to drag and lift of the various glider parts are shown in Tables II and III. In our simple analysis above (L/D)max occurs when the induced drag equals the friction and pressure drag combined. In the full analysis the induced drag fraction at (L/D)max is 43%, which is not far off the classical

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value of ½. Note that the fuselage, although of similar area to the tail, contributes less drag because of its higher Re . The gliding experiments were repeated with two new wings for the glider, both of the original area, one with four times the original aspect ratio (twice the span, half the chord) and one with ¼ of the aspect ratio (half the span, twice the chord). No measurable change in performance was observed, in accordance with equations 11 and 13. Acknowledgments I would like to thank my children Robert, Susanna and Christine, and my colleagues Rob Komar and Christian Nally for several patient hours operating a stop-watch and running up and down stairs to fetch the glider. Tables I. Glider specifications. Glider Specifications Mass 3.51 g Wing area 0.0070 m2 chord 36.5 mm aspect ratio 5.25 Fuselage area 0.0026 m2 length 150 mm Stabilizer area 0.0017 m2 chord 22 mm angle of attack -2.75° (at α = 0) Fin area 0.0006 m2 chord 20 mm Total surface area / wing area 1.7

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II. Drag contribution from the various parts of the glider at (L/D)max . Glider Part Fraction of Drag Wing (friction, pressure) 0.35 Stabilizer (friction, pressure) 0.11 Fuselage 0.06 Fin 0.04 Wing (induced) 0.40 Stabilizer (induced) 0.03 III. Lift contribution from the horizontal surfaces of the glider at (L/D)max. Glider Part Fraction of Lift Wing 0.88 Stabilizer 0.12 Figures 1. Force diagram for a balsa glider at constant velocity. 2. Variation of lift coefficient CL with angle of attack α for an ideal airfoil at high Re

and for a flat plate of A = 6 at Re ~ 104 (from Jones3). 3. Relationship between lift and induced drag resulting from airflow incident on a flat

plate. 4. Lift/drag ratios as a function of velocity for the glider showing measured data, and

calculations using both ideal flat plates and real plates with friction and pressure drag increased by 60%.

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L

Fin

W

D

Centre of Mass

Nose Weight

Stabilizer

Wing

Fuselage

Angle of Incidence, α Direction of Flight (θ to horizontal)

Force Diagram

Figure 1: Force diagram for a balsa glider at constant velocity.

Figure 2: Variation of lift coefficient CL with angle of attack α for an ideal airfoil at high Re and for a flat plate of A = 6 at Re ~ 104 (from Jones3).

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D i

Low Pressure

Flat Plate

L

α Air Flow

Total Pressure Force

High Pressure Figure 3: Relationship between lift and induced drag resulting from airflow incident on a flat plate.

Figure 4: Lift/drag ratios as a function of velocity for the glider showing measured data, and calculations using both ideal flat plates and real plates with friction and pressure drag increased by 60%.

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References

1 C. Waltham, Am. J. Phys. 65, 1082-1086 (1997) 2 F. M. White, Fluid Mechanics, (McGraw Hill, New York, 1979), p.427 3 B. Jones, Elements of Practical Aerodynamics, (John Wiley, New York, 1942), 3rd ed., p.21 4 J. D. Anderson Jr., A History of Aerodynamics, (Cambridge U.P., Cambridge, 1997), p.323 5 ibid, p.456