Historical Note: Aerodynamic Coefficients

In Section 1.5, we introduced the convention of expressing aerodynamic force in terms of an aerodynamic coefficient, such as

L = ^PocVISCl

D = pooVlSCD

where L and D are lift and drag, respectively, and Cl and Co are the lift coefficient and drag coefficient, respectively. This convention, expressed in the form shown above, dates from about 1920. But the use of some type of aerodynamic coefficients goes back much further. In this section, let us briefly trace the genealogy of aerodynamic coefficients. For more details, see the author’s recent book, A History of Aerodynamics and Its Impact on Flying Machines (Reference 62).

The first person to define and use aerodynamic force coefficients was Otto Lilien – thal, the famous German aviation pioneer at the end of the nineteenth century. Inter­ested in heavier-than-flight from his childhood, Lilienthal carried out the first defini­tive series of aerodynamic force measurements on cambered (curved) airfoil shapes using a whirling arm. His measurements were obtained over a period of 23 years, cul­minating in the publication of his book Der Vogelflug als Grundlage der Fliegekunst (Birdflight as the Basis of Aviation) in 1889. Many of the graphs in his book are plotted in the form that today we identify as a drag polar, i. e., a plot of drag coeffi­cient versus lift coefficient, with the different data points being measured at angles of attack ranging from below zero to 90°. Lilienthal had a degree in Mechanical Engineering, and his work reflected a technical professionalism greater than most at that time. Beginning in 1891, he put his research into practice by designing several gliders, and executing over 2000 successful glider flights before his untimely death in a crash on August 9, 1896. At the time of his death, Lilienthal was working on the design of an engine to power his machines. Had he lived, there is some conjecture that he would have beaten the Wright brothers in the race for the first heavier-than-air, piloted, powered flight.

In his book, Lilienthal introduced the following equations for the normal and axial forces, which he denoted by N and T, respectively (for normal and “tangential”)

N = 0.3r)FV2 T = 0A30FV2
[1.60]
and	[1.61]

where, in Lilienthal’s notation, F was the reference planform area of the wing in m2, V is the freestream velocity in m/s, and /V and T are in units of kilogram force (the force exerted on one kilogram of mass by gravity at sea level). The number 0.13 is Smeaton’s coefficient, a concept and quantity stemming from measurements made in the eighteenth century on flat plates oriented perpendicular to the flow. Smeaton’s coefficient is proportional to the density of the freestream; its use is archaic, and it went out of favor at the beginning of the twentieth century. By means of Equations

(1.60) and (1.61) Lilienthal introduced the “normal” and “tangential” coefficients, tj and в versus angle of attack. A copy of this table, reproduced in a paper by Octave Chanute in 1897, is shown in Figure 1.50. This became famous as the “Lilienthal Tables,” and was used by the Wright brothers for the design of their early gliders. It is proven in Reference 62 that Lilienthal did not use Equations (1.60) and (1.61) explicitly to reduce his experimental data to coefficient form, but rather determined his experimental values for i] and в by dividing the experimental measurements for N and T by his measured force on the wing at 90° angle of attack. In so doing, he divided out the influence of uncertainties in Smeaton’s coefficient and the veloc­ity, the former being particularly important because the classical value of Smeaton’s coefficient of 0.13 was in error by almost 40 percent. (See Reference 62 for more de­tails.) Nevertheless, we have Otto Lilienthal to thank for the concept of aerodynamic force coefficients, a tradition that has been followed in various modified forms to the present time.

Following on the heals of Lilienthal, Samuel Langley at the Smithsonian Institu­tion published whirling arm data for the resultant aerodynamic force R on a flat plate as a function of angle of attack, using the following equation:

R = kSV2F(ct) [1.62]

where S is the planform area, к is the more accurate value of Smeaton’s coefficient (explicitly measured by Langley on his whirling arm), and F (a ) was the correspond­ing force coefficient, a function of angle of attack.

The Wright brothers preferred to deal in terms of lift and drag, and used expres­sions patterned after Lilienthal and Langley to define lift and drag coefficients:

L = kSV2CL [1.63]

D = kSV2CD [1.64]

The Wrights were among the last to use expressions written explicitly in terms of Smeaton’s coefficient k. Gustave Eiffel in 1909 defined a “unit force coefficient” Ki as

R = KjSV2 [1.65]

In Equation (1.65), Smeaton’s coefficient is nowhere to be seen; it is buried in the direct measurement of A’,. (Eiffel, of Eiffel Tower fame, built a large wind tunnel in 1909, and for the next 14 years reigned as France’s leading aerodynamicist until his death in 1923.) After Eiffel’s work, Smeaton’s coefficient was never used in the aerodynamic literature—it was totally passe.

TABLE OF NORMAL AND TANGENTIAL PRESSURES

Deduced by Lilienthal from the diagrams on Plate VI., in his book “ Bird-flight as the Basis of the Flying Art”

a Angle,	* Normal.	a Tangential.	a Angle.	4 Normal.	» Tangential.
-9°………………………………………………	OjOOO	+ 01070	i6°……………	0.909	— 0X275
— 8°…………………………………………	охцо	+ OX267	17°………………………………………	0.915	— 0X273
-7°………………………………………………	0x280	+ ОЛ64	18°……………………………………….	0.919	— ОЛ7О
— 6°…………………………………………	a 120	+ 0uo6o	*9°……………………………………….	0921	— OX265
-s°………………………………..	0.160	+ ОЛ55	ao°………………………………………	0922	— OX259
-4°…………………………………………	0.200	+ OX249	ai°………………………………………..	0933	— OX253
-3°……………………………………………..	0.242	+ 0X243	aa°………………………………………	0934	— 0X247
– 2°…………………………………………….	0.286	+ 0X237	*3°……………………………………….	0.934	— 0X241
— 1°…………..	0.33a	+ OX23I	4°………………………………………	0923	— ОЛ36
0°………….	0.381	+ 0X224	4°………………………………………….	0^23	— 0X231
+ 1°…………………………………………..	0434	+ OX2I6	26°………………………………………	0.920	— 0X226
+ 3°…………………………………………	0489	+ 0Л08	27°…………..	0.918	— 0031
+ 3°……………	0.546	OdOOO	28°…………..	0.915	— 0x216
+ 4°………….	0.600	— OU007	39а…………..	Q.9I2	— 0012
+ 5°………….	0.650	— 0014	30°…………..	0910	—ox»8
+ 6°………….	0.696	— 0031	33°…………..	0906
+ 7°………….	0.737	— ftprf	35°…………..	0896	+0x210
+ 8°………….	0.771	—OX23S	40°…………..	0^90	+0x216
+ 9°………….	O.8OO	— 0043	45°…………..	ОІИМ	+0x220
up………	0.825	— 0050	s°°….	0888	+0x223
n°……….	0.846	— 0x258	55a…………..	0^90	+0x226
i2°………	0864	— 0x264	6o°…………..	0900	+ 0x228
«3°.	0879	— 0070	70°…………..	0.930	+0x230
•4*………	0Л91	—0x274	8o°………….	0960	+0.015
«5°…..	0901	— 0x276	90°………	1.000	0.000

Figure 1.50 The Lilienthal Table of normal and axial force coefficients. This is a facsimile of the actual table that was published by Octave Chanute in an article entitled "Sailing Flight," The Aeronautical Annual,

1 897, which was subsequently used by the Wright Brothers.

Gorrell and Martin, in wind tunnel tests carried out in 1917 at MIT on various airfoil shapes, adopted Eiffel’s approach, giving expressions for lift and drag:

L = K, AV2 [1.66]

D = KXAV2 [1.67]

where A denoted planform area and K, and Kx were the lift and drag coefficients, respectively. For a short period, the use of Kx and Kx became popular in the United States.

However, also by 1917 the density p began to appear explicitly in expressions for force coefficients. In NACA Technical Report No. 20, entitled “Aerodynamic Coefficients and Transformation Tables,” we find the following expression:

F = CpSV2

where F is the total force acting on the body, p is the freestream density, and C is the force coefficient, which was described as “an abstract number, varying for a given airfoil with its angle of incidence, independent of the choice of units, provided these are consistently used for all four quantities (F, p, S, and V).”

Finally, by the end of World War I, Ludwig Prandtl at Gottingen University in Germany established the nomenclature that is accepted as standard today. Prandtl was already famous by 1918 for his pioneering work on airfoil and wing aerodynamics, and for his conception and development of boundary layer theory. (See Section 5.8 for a biographical description of Prandtl.) Prandtl reasoned that the dynamic pressure, (he called it “dynamical pressure”), was well suited to describe aerodynamic force. In his 1921 English-language review of works performed at Gottingen before and during World War I (Reference 63), he wrote for the aerodynamic force,

W — cFq [1.68]

where W is the force, F is the area of the surface, q is the dynamic pressure, and c is a “pure number,” i. e., the force coefficient. It was only a short, quick step to express lift and drag as

L=qxSCL І1.69]

and D=qocSCo [1.70]

where CL and C n are the “pure numbers” referred to by Prandtl (i. e., the lift and drag coefficients). And this is the way it has been ever since.