The flight of a balsa gliderChris Waltham Citation: Am. J. Phys. 67, 620 (1999); doi: 10.1119/1.19334 View online: http://dx.doi.org/10.1119/1.19334 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v67/i7 Published by the American Association of Physics Teachers Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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The flight of a balsa gliderChris WalthamDepartment of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada

~Received 3 September 1998; accepted 11 November 1998!

A simple analysis is performed on the flight of a small balsa toy glider. All the basic features offlight have to be included in the calculation. Key differences between the flight of small objects likethe glider, and full-sized aircraft, are examined. Good agreement with experimental data is obtainedwhen only one parameter, the drag coefficient, is allowed to vary. The experimental drag coefficientis found to be within a factor of 2 of that obtained using the theory of ideal flat plates. ©1999

American Association of Physics Teachers.

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INTRODUCTION

Small gliders made of paper or balsa wood are frequeused in physics classes to demonstrate the balance of foand features of fluid mechanics such as Bernoulli’s theorOne immediate observation is that all gliders—balsapaper—have about the same glide slope, about 1 in 4, wcoincides neatly with the angle of the seating in a typilecture theatre. This seemingly innocuous feature is, hever, at odds with the behavior of larger models and fusized aircraft, the shape of whose wings critically affectsglide slope. The natural flying velocities of small gliders aalso very much alike: a few m/s. Full-sized aircraft shomuch greater variation, even if one only considers gliddesigned to carry a single person.1 The best competition glid-ers have long thin wings and can fly at 25 m/s with a glslope as good as 1 in 60; hang gliders have short stuwings and fly slower than 10 m/s with a glide slope steethan 1 in 10.

Presented here is a quantitative analysis of the flight osmall glider, a ‘‘Guillow Super-Ace’’~$1.99! made of sheetbalsa, of a sort which can be bought in any toy or mostore~see specifications in Table I!. Out of the analysis willappear broad features which are observed in the flight ohand-held gliders.

Flight is achieved by the interaction of a vehicle with tair surrounding it. As the aircraft moves through the air tflying surfaces deflect air downwards, creating a force whcan be resolved into components perpendicular to its mo~lift !, and parallel to its motion~‘‘induced’’ drag!. At thesame time the flow of air past the wings and body of the cis slowed by friction and changes in pressure caused byshape; this causes more resistance called friction and psure~or form! drag, respectively. A well-trimmed glider fliein a straight line at a constant speed, necessarily in a sligdownward direction, by balancing the forces of lift and drwith that of its weight. The flight path is determined by threlative sizes of the two types of drag force~Fig. 1!.

AERODYNAMICS AT LOW SPEEDS

The design criteria for a balsa glider are as follows. Tflight speed should be fast enough so it will not be unreasably affected by normal indoor turbulence~,1 m/s! but notso high as to be dangerous~.10 m/s!. These constraintsaffect the mass, which, as we will see below, is in the fgrams range for small hand-held aircraft, making balsa wor paper appropriate materials for construction.

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Any discussion of size and speed regimes in aerodynamleads us immediately to ask about the Reynolds numberas this will tell us what type of flow we are dealing with:

Re5rVx/m. ~1!

In the above equationr is the density of air,V the flowvelocity, m the viscosity, andx the relevant length scalefrequently the chord of the wing. The values ofr, V, andmrefer to values a large distance from the moving object.the case of air at STP, Re becomes in SI units

Re568 000Vx. ~2!

The Guillow Super Ace flies best at;4 m/s and it has amean chordc537 mm, so Re~for the wing! is ;10 000. TheReynolds number hierarchy of flying objects is as follows

~i! Insects: 102– 104;~ii ! Model Aircraft: 103– 106;~iii ! Birds: 104– 106;~iv! Aircraft: Human and solar powered 105– 106; Light

106– 107; Large 107– 108.

Aerodynamically this places our glider at the lower endthe model and bird regime and close to that of insectstakes us into the laminar flow regime, where drag is high alift poor,2 which occurs below Re;300 000. The comparisonwith birds is not far fetched; among the slowest birds arekinglets ~5.5 m/s! which have a wing loading (10 N/m2),which is approximately double that of our glider.3 Insects,however, fly by somewhat different means, relying on usteady airflow around their beating wings. Correct analyof insect flight has only recently come within the graspcomputational fluid dynamics.4

The lift force L is understood in terms of speed, air desity, plan wing area~S!, and a coefficient of liftCL . Theexpression for lift has the following form; for flight at constant velocity at angleu to the horizontal, the lift of the wingmust equal the component of the weight of the aircraftWperpendicular to the flight path~Fig. 1!:

L5 12CLrV2S5W cosu. ~3!

For a wing with a thin airfoil section, aspect ratio~span/meanchord! A, at incident anglea, with an attached boundarlayer,5

CL5l~Re!•2p sina

112/A. ~4!

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The value ofl~Re! approaches unity for Re.106, but for ourflight regime (Re'104), it rather less. Equation~4! is alsotrue for a flat plate, but for a much smaller range of angthan for more appropriately shaped airfoils. Abovea;10°,CL starts to drop below the ideal value, and above;14°, itdrops precipitously~i.e., the wing stalls!. Hard data are rareflat plate wings have not been very interesting since HorPhillips, in the last years of the 19th century, found cabered ones to be better. Bradley Jones’ textbook6 of 1942shows a useful plot for a flat plate withA56 ~very close tothe glider’s value of 5.25! at about the right Re~Fig. 2!. Thiscan be parametrized with a fair degree of accuracy usingexponential to describe the stalling behavior:

CL54.0 sina20.0037 exp~a/3.5°!. ~5!

Similarly for the drag forceD,

D5 12CDrV2S. ~6!

The drag coefficientCD is made of up of three terms, friction, pressure drag, and induced drag. The first, for flat shparallel to the fluid flow, in the laminar region, is given bthe expression first derived by in 1908 by Blasius:7

CD, f52.66 Re20.5 for Re,;500 000. ~7!

The second term, pressure drag~coefficientCD,P!, is zero forideal flat sheets. We handle any deviation from idealitymultiplying the friction drag coefficient by a factorf.

Table I. Glider specifications.

Glider specifications

Mass 3.51 gWing area 0.0070 m2

chord 36.5 mmaspect ratio 5.25

Fuselage area 0.0026 m2

length 150 mmStabilizer area 0.0017 m2

chord 22 mmangle of attack 22.75° ~at a50!

Fin area 0.0006 m2

chord 20 mmTotal surface area/wing area 1.7

Fig. 1. Force diagram for a balsa glider at constant velocity.

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The third term, induced drag, can be handled simplyflat sheets, because the total force on the sheet is a resua pressure difference above and below the sheet, and soto be normal to the surface8 ~Fig. 3!. The diagram shows thenet force as passing through the center of the sheet. Inthe force intersects the sheet somewhat forward of this pas can be observed by trying to ‘‘fly’’ a wing made of a pieof card; it spins about its long axis with the leading edrising. However, this position shift has no effect on therection of the induced drag. The induced drag can be inferfrom the lift, and has the same dependency as lift onr, V,andS:

Di5L tana; CD,i5CL tana. ~8!

Thus for smalla, CD,i is approximately proportional toCL2,

and therefore only slightly dependent on the aspect ratioA.For good airfoils at high Re, the expression for induced dis as follows:

CD,i'CL

2

pA. ~9!

Thus the dependence onA is much stronger, and this favorlong, thin wings for commercial aircraft~A;8 for wide-body jets! and full-sized gliders~A up to ;30!.

In all cases, the total drag is given by

D5 12CDrV2S5 1

2~CD, f1CD,P1CD,i !rV2S. ~10!

Fig. 2. Variation of lift coefficientCL with angle of attacka for an idealairfoil at high Re and for a flat plate ofA56 at Re;104 ~from Ref. 4!.

Fig. 3. Relationship between lift and induced drag resulting from airflincident on a flat plate.

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We can use these equations to optimize whatever flight qity we wish, in particular range and therefore glide angRange is how far an aircraft can fly with a given amountenergy~chemical energy in the fuel for a powered aircrapotential energy for a glider!. The glide angle~see Fig. 1! isu, the angle the flight direction makes to the horizontal, ait is related to the ratio of lift to drag. This ratio is atmaximum, (L/D)max, when the angle is at a minimum,u0 .For small angles of attack~tana'sina! we can estimate(L/D)max using Eq.~7!:

L/D5CL /CD'l•2p sina/~112/A!

l•2p sin2 a/~112/A!1CD, f. ~11!

This can be maximized by differentiating with respectsina. The maximum occurs when the two terms in the dnominator are equal, i.e., friction drag equals induced dr

~L/D !max5~CL /CD!max'Al•p/~112/A!

2CD, f. ~12!

Note the very weak dependence of (L/D)max on aspect ratio,velocity, and chord, from which we can expect a fairly unversal value for all hand-held gliders. We can see what thfrom an estimate of the performance of our particular glidwhich from experiment flies best at Re'10 000. The Blasiusformula gives the drag coefficient, which is Eq.~7! multi-plied by the total surface area divided by the wing area~afactor 1.70 in this case!. The aspect ratioA is 5.25 and fromthe parametrization of Jones’ data@Eq. ~5!# l'0.9. Thisyields (L/D)max'5. Using this result and Eqs.~3!, ~4! and~6! givesn0'5 m/s. This velocity depends on the wing loaing (W/S), the wing chord~c!, and sinu0@51/(L/D)max#:

n03/2}

W sinu0c1/2

S. ~13!

Hence we can see that for a hand-held glider of a few grathe natural flying speed will not deviate from the rangeseveral m/s. This simple analysis comes close to our msured values of (L/D)max54.25 atn053.7 m/s, and the discrepancy indicates that the friction drag is greater thanideal flat sheets, and/or that the pressure drag is nonzer‘‘normal’’ aerodynamics, i.e., good airfoils at high Re, th

Fig. 4. Lift/drag ratios as a function of velocity for the glider showinmeasured data, and calculations using both ideal flat plates and real pwith friction and pressure drag increased by 60%.

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expression for (L/D)max is as follows. Note again the mucstronger dependence on aspect ratio:

~L/D !max5~CL /CD!max'1

2ApA/~112/A!

CD, f1CD,P. ~14!

Thus to achieve an (L/D)max of 60, the Nimbus-3 full-sizecompetition glider1 has long thin wings (A530) and an ex-ceptionally low drag coefficient~friction plus pressure! of0.006. This value is1

20 of that of our model glider, as we wilsee below.

DATA AND ANALYSIS

The experiment was performed in a large, steep, emlecture theatre. The glider was thrown many times and geffort was made to launch it at about the right angle aspeed to give it a linear trajectory. The flight was timed athe impact point was noted so thatn andu could be obtainedusing a measuring tape and a theodolite. The flight path25 m long. The position of the wing in the fuselage wvaried to optimize the glide slope. The process was not eand only a small fraction of the flights were straight enouto give good data; this was particularly true below 3.3 mand above 5.5 m/s. The results are shown in Fig. 4.

Modeling the flight of the glider required a straightforwaiterative calculation. There was only one free parameter,drag factorf, which was expected to be close to unity. Tsteps were as follows:

~i! Start with angle of attacka; allow it to vary from 0 to10° in 0.1° steps.

~ii ! Calculate the lift at required at this angle,W cosu,~start withu590° for a50!.

~iii ! Guess a starting value for the total drag coefficieCD .

~iv! Determine Re values for the wing, fuselage, stabilizand fin.

~v! Calculate the drag for each member asf3(2.66 Re20.5).

~vi! Find the induced drag.~vii ! Add all the drags to find the total.~viii ! Find (L/D) for each value ofa, and hence findu.

Feed to next value ofa.

tes

Table II. Drag contribution from the various parts of the glider at (L/D)max.

Glider part Fraction of drag

Wing ~friction, pressure! 0.35Stabilizer~friction, pressure! 0.11Fuselage 0.06Fin 0.04Wing ~induced! 0.40Stabilizer~induced! 0.03

Table III. Lift contribution from the horizontal surfaces of the glider(L/D)max.

Glider part Fraction of lift

Wing 0.88Stabilizer 0.12

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When the calculation was done for all values ofa:

~i! Read off (L/D)max and findn0 .~ii ! Vary f to reproduce the experimental values.~iii ! Check that the results are independent of the choic

initial CD .

One implicit simplification should be noted: we have asumed, without mathematical justification, that there isinterference between the components of the glider, whhave been treated separately. The wings will have a smeffect on the drag of the fuselage, as they will on the airflaround the tail.

The result for the drag factor isf 51.6860.08 (x2/d f53.9) using all six data points, andf 51.5860.07 (x2/d f51.7) if we ignore the two extreme velocities~at which theglider was hard to fly straight!. As the exact value is not opivotal importance, let us sayf 51.660.2 and note that it isclose to unity. There will be an increase in drag causedblunt leading edges and rough surfaces. There will also bdecreasein drag due to the separated flow over the top ofwing suffering less friction than the Blasius formula@Eq. ~7!#would indicate.8 Determining which effect is greater is beyond simple theory. Our analysis withf 51.6 gives(L/D)max54.06 atn054.095 m/s~Re510 000! anda56.5°,with a total drag coefficient,CD50.12. The contributions todrag and lift of the various glider parts are shown in Tableand III. In our simple analysis above (L/D)max occurs whenthe induced drag equals the friction and pressure drag cbined. In the full analysis the induced drag fraction

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(L/D)max is 43%, which is not far off the classical value o12 . Note that the fuselage, although of similar area to the tcontributes less drag because of its higher Re.

The gliding experiments were repeated with two newings for the glider, both of the original area, one with fotimes the original aspect ratio~twice the span, half the chord!and one with1

4 of the aspect ratio~half the span, twice thechord!. No measurable change in performance was obserin accordance with Eqs.~11! and ~13!.

ACKNOWLEDGMENTS

I would like to thank my children Robert, Susanna, aChristine, and my colleagues Rob Komar and ChristNally for several patient hours operating a stop-watch arunning up and down stairs to fetch the glider.

1P. P. Wegener,What Makes Airplanes Fly?~Springer, New York, 1991!,p. 169.

2C. Waltham, ‘‘Scaling in model aircraft,’’ Am. J. Phys.65, 1082–1086~1997!.

3H. Tennekes,The Simple Science of Flight~MIT, Cambridge, MA, 1997!,p. 127.

4See, for example, C. P. Ellington, C. van den Berg, and P. A. Willm‘‘Leading Edge Vortices in Insect Flight,’’ Nature~London! 384, 626–630~1996!.

5F. M. White,Fluid Mechanics~McGraw–Hill, New York, 1979!, p. 427.6B. Jones,Elements of Practical Aerodynamics~Wiley, New York, 1942!,3rd ed., p. 21.

7J. D. Anderson, Jr.,A History of Aerodynamics~Cambridge U.P., Cam-bridge, 1997!, p. 323.

8Ibid, p. 456.

THE LIFE OF AN ASSISTANT PROFESSOR

Your salary as an assistant professor, as for all professors, will not only reflect your seniority,or in this case your lack of it, but also your success at bringing in outside money. Since you arejust starting out, you will have had no such success. Therefore your salary will be miserly to poor.If you are such an exciting prospect that you have managed to land an assistant professorship at amajor private university with a fancy reputation, then your salary will be even worse. Suchuniversities expect you to accept lower pay in return for the snob appeal of their name on yourresume´. They also offer significantly reduced, if any, opportunity for promotion to tenure, on theperhaps correct assumption that their name is worth more to you than job security.

Peter J. Feibelman,A Ph.D. Is Not Enough—A Guide to Survival in Science~Addison–Wesley, Reading, MA, 1993!, p. 60.

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