Aerodynamicists control the flow field by defining the configuration geometry, and they are always interested in geometric shapes that could possibly be useful in design. Though this appendix provides the detailed definition of many of the classic shapes frequently specified in aerodynamics, it is not encyclopedic.

We discuss airfoil geometry in section A.1, followed by classic bodies of revolution in section A.2. In section A.3, we present a class of cross-sectional shapes that can be used to develop more realistic models than bodies of revolution. Planform properties of interest in aerodynamic analysis are presented in section A.4. Sources for information on conical camber and 3D wing geometry are highlighted in sections A.5 and A.6, respectively.

A.1 Airfoil Geometry

This section is divided into two subsections. In section A.1.1, we discuss NACA airfoils, which are generally based on simple geometrical definitions of the section shapes described by equations. Most modern airfoils are defined only by a table of coordinates, not equations. A long list of the tabulated airfoils is presented in section A.1.2. A number of references are included to allow the reader to study both the older NACA literature and new airfoil design ideas. As a whole, this literature provides a means of obtaining a rather complete understanding of the ways in which airfoils can be shaped to obtain desired performance characteristics.

A.1.1 The NACA Airfoils

The NACA airfoils were designed between 1929 and 1947 under the direction of Eastman Jacobs at the NACA’s Langley Field Laboratory. Most of the airfoils were based on simple geometrical descriptions of the section shape, although the 6 and 6A series were developed using theoretical analysis and don’t have simple shape definitions. Although a new generation of airfoils has emerged as a result of improved understanding of airfoil performance and the ability to design new airfoils using computational methods, the NACA airfoils are still useful in many aerodynamic design applications.

The NACA airfoils are constructed by combining a thickness envelope with a camber or mean line. The equations that describe these shapes are

$$\begin{array}{rl}& x_u=x-y_t(x)\sin \theta \\ & y_u=y_c(x)+y_t(x)\cos \theta \end{array}\text{(A-1)}$$

and

$$\begin{array}{rl}& x_l=x+y_t(x)\sin \theta \\ & y_l=y_c(x)-y_t(x)\cos \theta \end{array}\text{(A-2)}$$

where y_t(x) is the thickness function, y_c(x) is the camber line function, and

$$\theta ={\tan }^{-1}\left(\frac{d{y}_c}{dx}\right)\text{(A-3)}$$

is the camber line slope. It is not unusual to neglect the camber line slope terms in equations (A-1) and (A-3). This simplifies the equations and makes the reverse problem of extracting the thickness envelope and mean line for a given airfoil quite straightforward.

The primary reference for all the NACA subsonic airfoil studies is 1959’s Theory of Wing Sections by Abbott and von Doenhoff (Dover Publications, Inc.). A brief history of the evolution of the NACA Airfoils is presented in the table below. We also provide references to the development of the NASA advanced airfoils from 1966 to approximately 1977. The original NACA Reports cited below are now available online as PDF files through NASA and are well worth reading.

Table A-1: Evolution of the NACA airfoils. (*Additional section data for NACA 6- and 6A-series airfoils is in NASA Technical Report R-84 authored by Patterson and Braslow and published in 1958.)

These airfoils, regardless of the combination of camber lines and thickness envelopes, can be constructed using program FOILGEN, described in appendix E.

Historical accounts of the NACA airfoil program are contained in the following sources:

Abbott, I. H., “Airfoils,” in “Evolution of Aircraft Wing Design,” AIAA Dayton Section Symposium, Mar. 1980, AIAA Paper 80-3033. https://doi.org/10.2514/6.1980-3033

Jones, R. T., “Recollections From an Earlier Period in American Aeronautics,” Annual Review of Fluid Mechanics, Vol. 9, pp. 1–11, 1977. https://doi.org/10.1146/annurev.fl.09.010177.000245

NASA has published two reports describing computer programs that produce the NACA airfoil ordinates:

Ladson, C. L., and Brooks, C. W., Jr., “Development of a Computer Program to Obtain Ordinates for the NACA 4-Digit, 4-Digit Modified, 5-Digit, and 16-Series Airfoils,” NASA TM X-3284, Nov. 1975. https://ntrs.nasa.gov/citations/19760003945

Ladson, C. L., and Brooks, C. W., Jr., “Development of a Computer Program to Obtain Ordinates for the NACA 6- and 6A-Series Airfoils,” NASA TM X-3069, Sept. 1974. https://ntrs.nasa.gov/citations/19740025318 (Note: This program is included in the utility programs described in App. E, as LADSON. It is not extremely accurate for sections less than 6% thick or greater than 15% thick.)

An extensive and excellent survey of the older airfoils is contained in:

Riegels, F. W., Airfoil Sections (English-language version), Butterworths, London, 1961.

NASA supercritical airfoil development is described in the following references:

Whitcomb, R. T., “Review of NASA Supercritical Airfoils,” ICAS Paper 74-10, 9th Congress of the International Council of the Aeronautical Sciences, Haifa, Israel, Aug. 25-30, 1974. https://www.icas.org/icas_archive/ICAS1974/Page%208%20Whitcomb.pdf

Harris, C. D., “NASA Supercritical Airfoils,” NASA TP-2969, Mar. 1990. https://ntrs.nasa.gov/citations/19900007394

Becker, J. V., “The High-Speed Airfoil Program,” in The High Speed Frontier, NASA SP-445, 1980. https://www.nasa.gov/wp-content/uploads/2024/01/sp-445.pdf

A.1.1.1 The NACA 4-Digit Airfoil

The numbering system for these airfoils is defined by NACA as MPXX, where

XX is the maximum thickness, t/c, in percent chord,

M is the maximum value of the mean line in hundredths of chord, and

P is the chordwise position of the maximum camber in tenths of the chord.

Note that although the numbering system implies integer values, the equations can provide 4-digit foils for arbitrary values of M, P, and XX.

An example: NACA 2412

It is a 12-percent-thick airfoil with a max value of the camber line of 0.02, at x/c = 0.4.

The NACA 4-digit thickness distribution is given by

$$\frac{y_t}{c}=\left(\frac{t}{c}\right)[a_0\sqrt{x/c}-a_1(x/c)-a_2(x/c{)}^2+a_3(x/c{)}^3-a_4(x/c{)}^4]\text{(A-4)}$$

where

a₀ = 1.4845;  a₁ = 0.6300;  a₂ = 1.7580;  a₃ = 1.4215; and a₄ = 0.5075.

The maximum thickness occurs at x/c = 0.30, and the leading edge radius is

$$\left(\frac{r_{LE}}{c}\right)=1.1019{\left(\frac{t}{c}\right)}^2\text{.}\text{(A-5)}$$

The included angle of the trailing edge is

$${\delta }_{TE}=2{\tan }^{-1}\left\{1.16925\left(\frac{t}{c}\right)\right\}\text{.}\text{(A-6)}$$

It is important to realize that the airfoil has a finite thickness at the trailing edge. The equation (A-4) definition of the thickness distribution results in a small but finite trailing edge thickness. Many computational methods require a zero-thickness trailing edge, and the coefficient definitions are frequently modified to produce a zero-thickness trailing edge. This can lead to the wrong value of drag from calculations. Van Dam^[1] cites a value of wave drag at M = 0.78, which is 15.3% too high for the 0012 airfoil modified to zero trailing edge thickness!

The camber line is given by

$$\begin{array}{rl}\frac{y_c}{c}& =\frac{M}{(p{)}^2}[2P(x/c)-(x/c{)}^2]\\ \frac{d{y}_c}{dx}& =\frac{2M}{(p{)}^2}(P-(x/c))\end{array}\}\left(\frac{x}{c}\right)\text{less than}P\text{(A-7)}$$

and

$$\begin{array}{rl}\frac{y_c}{c}& =\frac{M}{(1-P{)}^2}[1-2P+2P(x/c)-(x/c{)}^2]\\ \frac{d{y}_c}{dx}& =\frac{2M}{(1-P{)}^2}(P-(x/c))\end{array}\}\left(\frac{x}{c}\right)\ge P\text{.}\text{(A-8)}$$

The camber line slope is found from equation (A-3) using equations (A-7) and (A-8), and the upper and lower surface ordinates resulting from the combination of thickness and camber are then computed using equations equations (A-1) and (A-2).

A.1.1.2 The NACA 5-Digit Airfoil

This airfoil is an extension of the 4-digit series that provides additional camber lines. The numbering system for these airfoils is defined by NACA as LPQXX, where:

XX is the maximum thickness, t/c, in percent chord,

L is the amount of camber; the design lift coefficient is (3/2) L, in tenths,

P is the designator for the position of maximum camber, x_f, where x_f = P/2, and P is given in tenths of the chord, and

Q = 0 standard 5-digit foil camber, and Q = 1 “reflexed” camber.

An example: NACA 23012

For a 12-percent-thick airfoil, the design lift coefficient is 0.3 [(3/2)2 in tenths], the position of max camber is located at x/c = 0.15, and the “standard” 5-digit foil camber line is used.

The thickness distribution is the same as that of the NACA 4-digit airfoil described above in equation (A-4).

The standard 5-digit series camber line is given by

$$\begin{array}{rl}\frac{y_c}{c}& =\frac{K_1}{6}[(x/c{)}^3-3m(x/c{)}^2+m^2(3-m)(x/c)]\\ \frac{d{y}_c}{dx}& =\frac{K_1}{6}[3(x/c{)}^2-6m(x/c)+m^2(3-m)]\end{array}\}0\le (x/c)\le m\text{(A-9)}$$

and

$$\begin{array}{rl}\frac{y_c}{c}& =\frac{K_1}{6}{m}^3[1-(x/c)]\\ \frac{d{y}_c}{dx}& =-\frac{K_1}{6}{m}^3\end{array}\}m<(x/c)\le 1\text{(A-10)}$$

where m is not the position of maximum camber, but is related to the maximum camber position by

$$x_f=m(1-\sqrt{\frac{m}{3}})\text{,}\text{(A-11)}$$

and m is found from a simple fixed-point iteration for a given x_f. To avoid the leading edge singularity for a prescribed C_li and m, K₁ is defined as

$$K_1=\frac{6C_{l_i}}{Q}\text{(A-12)}$$

where

$$Q=\frac{3m-7m^2+8m^3-4m^4}{\sqrt{m(1-m)}}-\frac{3}{2}(1-2m)[\frac{\pi }{2}-{\sin }^{-1}(1-2m)]\text{.}\text{(A-13)}$$

Note that K₁ is a linear function of C_li and the K₁’s were originally tabulated for C_li = 0.3. The tabulated K₁’s are multiplied by (C_li/0.3) to get values at other C_li. To compute the camber line, the values of Q and K₁ must be determined. In some cases, the computed values of K₁ and Q differ slightly from the official tabulated values (remember these were computed in the 1930s). The tabulated values should be used to reproduce the official ordinates. Table A-2 illustrates the differences.

Table A-2: Difference between computed and tabulated values

Once the camber line parameters are chosen, the airfoil is constructed using the equations given above.

Camber lines designed to produce zero pitching moment

The reflexed mean line equations were derived to produce zero pitching moment about the quarter chord:

$$\frac{y_c}{c}=\frac{K_1}{6}[\{(x/c)-m{\}}^3-\frac{K_2}{K_1}(1-m{)}^3(x/c)-m^3(x/c)+m^3]0\le (x/c)\le m\text{(A-14)}$$

$$=\frac{K_1}{6}[\frac{K_2}{K_1}\{(x/c)-m{\}}^3-\frac{K_2}{K_1}(1-m{)}^3(x/c)-m^3(x/c)+m^3]0<(x/c)\le 1\text{(A-15)}$$

where

$$\frac{K_2}{K_1}=\frac{3{(m-x_f)}^2-m^3}{(1-m{)}^3}\text{.}\text{(A-16)}$$

The parameters are defined as follows:

1. given x_f , find m to give C_mc/4 = 0 from thin airfoil theory; and
2. given x_f and m, calculate K₁ to give C_li = 0.3.

The tabulated values for these camber lines are described in table A-3 below.

Table A-3: Tabulated values for camber lines designed to produce zero pitching moment

A.1.1.3 The NACA Modified 4-Digit Airfoil

This airfoil is an extension of the 4-digit series that allows for a variation of leading edge radius and location of maximum thickness. The numbering system is defined by NACA as MPXX-IT, where MPXX is the standard 4-digit designation and the IT appended at the end describes the modification to the thickness distribution. They are defined as:

I = designation of the leading edge radius, and

T = chordwise position of maximum thickness in tenths of chord.

These are implemented as:

$$\frac{r_{le}}{c}=1.1019{(\frac{I}{6}\cdot \frac{t}{c})}^2\text{for I}\le 8\text{(A-17)}$$

and

$$\frac{r_{\text{le}}}{c}=3\times 1.1019{\left(\frac{t}{c}\right)}^2\text{for I}=9\text{.}\text{(A-18)}$$

I = 6 produces the leading edge radius of the standard 4-digit airfoils.

One example of the modified 4-digit airfoil is NACA 0012-74. It is an uncambered 12-percent-thick airfoil, with a maximum thickness at x/c = 0.40 and a leading edge radius of 0.0216, which is 36% larger than the standard 4-digit value.

The NACA 16-series is a special case of the modified 4-digit airfoil with a leading edge radius index of I = 4 and the maximum thickness located at x/c = 0.5 (T = 5). As an example, the NACA 16-012 is equivalent to an NACA 0012-45.

The thickness distribution is given by

$$\frac{y_t}{c}=5\left(\frac{t}{c}\right)[a_0\sqrt{\frac{x}{c}}+a_1\left(\frac{x}{c}\right)+a_2{\left(\frac{x}{c}\right)}^2+a_3{\left(\frac{x}{c}\right)}^3]0\text{less than}\frac{x}{c}\text{less than}T\text{(A-19)}$$

and

$$\frac{y_t}{c}=5\left(\frac{t}{c}\right)[.002+d_1(1-\frac{x}{c})+d_2{(1-\frac{x}{c})}^2+d_3{(1-\frac{x}{c})}^3]T\text{less than}\frac{x}{c}\le 1\text{(A-20)}$$

The coefficients are determined by solving for the d’s first, based on the trailing edge slope and the condition of maximum thickness at x/c = T. Once these coefficients are found, the a’s are found by relating a₀ to the specified leading edge radius, the maximum thickness at x/c = T, and the condition of continuity of curvature at x/c = T. These constants are all determined for t/c = 0.2 and then scaled to other t/c values by multiplying by 5(t/c). The value of d₁ controls the trailing edge slope and was originally selected to avoid reversals of curvature. In addition to the tabulated values, Riegels^[2] has provided an interpolation formula. The official (tabulated) and Riegels approximate values of d₁ are given in table A-4.

Table A-4: Riegels approximate values of d₁

In this table, the Riegels approximate d₁ is defined by

$$d_1\cong \frac{(2.24-5.42T+12.3T^2)}{10(1-0.878T)}\text{.}\text{(A-21)}$$

Once the value of d₁ is known, d₂ and d₃ are found from the relations given by Riegels:

$$d_2=\frac{0.294-2(1-T)d_1}{(1-T{)}^2}\text{(A-22)}$$

and

$$d_3=\frac{-0.196+(1-T)d_1}{(1-T{)}^3}\text{.}\text{(A-23)}$$

With the d’s determined, the a’s can be found. a₀ is based on the leading edge radius:

$$a_0=0.296904\cdot {\chi }_{LE}\text{(A-24)}$$

where

$$\begin{array}{rlrl}{\chi }_{LE}& =\frac{I}{6}& & \text{for}I\le 8\\ & =10.3933& & \text{for}I=9\end{array}\text{.}\text{(A-25)}$$

Define the following:

$${\rho }_1=\left(\frac{1}{5}\right)\frac{(1-T{)}^2}{[0.588-2d_1(1-T)]}\text{,}\text{(A-26)}$$

then the rest of the a’s can be found from:

$$a_1=\frac{0.3}{T}-\frac{15}{8}\cdot \frac{a_0}{\sqrt{T}}-\frac{T}{10{\rho }_1}\text{(A-27)}$$

$$a_2=-\frac{0.3}{T^2}+\frac{5}{4}\cdot \frac{a_0}{T^{3/2}}+\frac{1}{5{\rho }_1}\text{(A-28)}$$

$$a_3=\frac{0.1}{T^3}-\frac{0.375a_0}{T^{5/2}}-\frac{1}{10{\rho }_{1}T}\text{.}\text{(A-29)}$$

The camber lines are identical to the standard 4-digit airfoils described previously. The upper and lower ordinates are then computed using the standard equations.

A.1.1.4 The NACA 6 and 6A-Series Mean Lines

For these airfoils, only the mean lines have analytical definitions. The thickness distributions are the result of numerical methods which produced tabulated coordinates. In addition to the values tabulated in the NACA reports, the closest approximation for the thickness distributions is available in the program LADSON (see appendix E).

The 6-series mean lines were designed using thin airfoil theory to produce a constant loading from the leading edge back to x/c = a, after which the loading decreases linearly to zero at the trailing edge. Theoretically, the loading at the leading edge must be either zero or infinite within the context of thin airfoil theory analysis. The violation of the theory by the assumed finite leading edge loading is reflected by the presence of a weak singularity in the mean line at the leading edge, where the camber line has an infinite slope. Therefore, according to Abbott and von Doenhoff,^[3] the 6-series airfoils were constructed by holding the slope of the mean line constant in front of x/c = 0.005, with the value at that point. For round leading edges, the camber line values are essentially not used at points ahead of the origin of the leading edge radius. The theory is discussed by Abbott and von Doenhoff on pages 73–75, 113, and 120. Tabulated values are contained on pages 394–405. The derivation of this mean line is a good exercise in thin airfoil theory. By simply adding various mean lines together, other load distributions can be constructed.

To quote from Abbott and von Doenhoff:

The NACA 6-series wing sections are usually designated by a six-digit number together with a statement showing the type of mean line used. For example, in the designation NACA 65,3-218, a = 0.5, the 6 is the series designation. The 5 denotes the chordwise position of minimum pressure in tenths of the chord behind the leading edge for the basic symmetrical section at zero lift. The 3 following the comma (sometimes this is a subscript or in parenthesis) gives the range of lift coefficient in tenths above and below the design lift coefficient in which favorable pressure gradients exist on both surfaces. The 2 following the dash gives the design lift coefficient in tenths. The last two digits indicate the thickness of the wing section in percent chord. The designation a = 0.5 shows the type of mean line used. When the mean line is not given, it is understood that the uniform-load mean line (a = 1.0) has been used.

The 6A series airfoils employed an empirical modification of the a = 0.8 camber line to allow the airfoil to be constructed of nearly straight line segments near the trailing edge. This camber line is described by Loftin in NACA R-903.

Basic camber line equations

When a = 1 (uniform loading along the entire chord),

$$\frac{y}{c}=-\frac{C_{l_i}}{4\pi }[(1-\frac{x}{c})\ln (1-\frac{x}{c})+\frac{x}{c}\ln \left(\frac{x}{c}\right)]\text{(A-30)}$$

and

$$\frac{dy}{dx}=\frac{C_{l_i}}{4\pi }[\ln (1-\frac{x}{c})-\ln \left(\frac{x}{c}\right)]\text{(A-31)}$$

where C_li is the “ideal” or design lift coefficient, which occurs at zero angle of attack.

For a < 1,

$$\frac{y}{c}=\frac{C_{l_i}}{2\pi (1+a)}\left\{\begin{array}{c}\frac{1}{1-a}\left[\begin{array}{c}\frac{1}{2}{(a-\frac{x}{c})}^2\ln |a-\frac{x}{c}|-\frac{1}{2}{(1-\frac{x}{c})}^2\ln (1-\frac{x}{c})\\ +\frac{1}{4}{(1-\frac{x}{c})}^2-\frac{1}{4}{(a-\frac{x}{c})}^2\end{array}\right]\\ -\frac{x}{c}\ln \left(\frac{x}{c}\right)+g-h\frac{x}{c}\end{array}\right\}\text{(A-32)}$$

with

$$g=\frac{-1}{(1-a)}[a^2(\frac{1}{2}\ln a-\frac{1}{4})+\frac{1}{4}]\text{,}\text{(A-33)}$$

$$h=(1-a)[\frac{1}{2}\ln (1-a)-\frac{1}{4}]+g\text{,}\text{(A-34)}$$

and

$$\frac{dy}{dx}=\frac{C_{l_i}}{2\pi (1+a)}\{\frac{1}{1-a}[(1-\frac{x}{c})\ln (1-\frac{x}{c})-(a-\frac{x}{c})\ln (a-\frac{x}{c})]-\ln \left(\frac{x}{c}\right)-1-h\}\text{.}\text{(A-35)}$$

The associated angle of attack is

$${\alpha }_i=\frac{C_{l_i}h}{2\pi (1+a)}\text{(A-36)}$$

a = .8 (modified), the 6A-series mean line

For 0 < x/c < .87437, use the basic a = .8 camber line, but with a modified value of the ideal lift coefficient, C_limod = C_li/1.0209. For .87437 < x/c < 1, use the linear equation:

$$\frac{y_c/c}{C_{l_i}}=0.0302164-0.245209(\frac{x}{c}-0.87437)\text{(A-37)}$$

and

$$\frac{dy}{dx}=-0.245209C_{l_i}\text{.}\text{(A-38)}$$

Note that at x/c = 1, the foregoing approximate relation gives y/c = -0.000589, indicating an α shift of 0.034° for C_li = 1.

A.1.1.5  Other airfoil definition procedures

Interest in defining airfoils by a small number of parameters for use in numerical optimization has led to several recent proposed parametric representations that might be useful. In particular, works by Verhoff et al.^[4] and Stookesberry et al.^[5] use Chebyshev functions to obtain functions with can represent very general airfoil shapes requiring five to twenty coefficients.

Ventkataraman.^[6],[7] employed another approach that uses Bezier methods frequently featured in CAD surface representation software. This approach uses fourteen design variables to represent the airfoil.  Sadrehaghighi et al.^[8] have used a similar approach based on nonuniform rational B-splines (NURBS).

A.1.2 Tabulated Airfoil Definition and the Airfoil Library

Most modern airfoils are not described by equations; instead, they are defined by a table of coordinates. Frequently, these coordinates are the results of a computational aerodynamic design program, and simple algebraic formulas cannot be used to define the shape. This was the case with the NACA 6-series airfoils described above.

The following table provides a list of the tabulated airfoils currently available that you can obtain as a ZIP file by accessing https://archive.aoe.vt.edu/mason/Mason_f/foils.zip. The subsequent tables provide a guide to these airfoils. A standard 2F10 format (the Jameson input format) is used with each set of coordinates. The values are given from leading edge to trailing edge, top followed by bottom.

Table A-5: Airfoil library files

The coordinates for the NACA 64A410 airfoil, widely used as a test case, are found on page 356 of the book by Abbott and von Doenhoff; be aware that this page contains some typos. In particular, the value of y at x = 7.5 is supposed to be 2.805. Also, it is obvious that the x value between 30 and 40 is 35. Finally, the table is mislabeled. The y typo was responsible for the widely circulated story that the coordinates contained in Abbott and von Doenhoff were not accurate enough to be used in modern computational airfoil codes.

It is also noteworthy that coordinates for the NASA Supercritical Airfoils were typed in by students when NASA TP-2969 was issued. The coordinates are exactly the values printed in the TP. It turns out that those values were developed by taking the original set of coordinates and enhancing them to 100 points upper and lower using straight-line interpolation. As such, they are not really usable. Our experience is that we can identify the original values and weed out the straight-line interpolated values. This works.

Many other airfoils are available on the World Wide Web. In particular, the Applied Aerodynamics group at the University of Illinois, under the direction of Prof. Michael Selig, has established a massive online source for airfoil definitions and includes data from wind tunnel tests on the airfoils available at https://m-selig.ae.illinois.edu/ads/coord_database.html. Their focus is directed toward airfoils designed for low speeds and low Reynolds numbers. Finally, Richard Eppler has published an entire book of his airfoils, Airfoil Design and Data, published by Springer-Verlag in 1990.

The NASA low- and medium-speed airfoils, including natural laminar flow airfoils, are listed in the table below.

Table A-6: NASA low speed, medium speed, and natural laminar flow airfoil

The NASA Phase 2 supercritical airfoils are listed in the following table.

Table A-7: NASA supercritical airfoils—Phase 2, from NASA TP 2969, March 1990, by Charles D. Harris

Several transonic airfoils were developed at New York University by a group led by Paul Garabedian. The following table provides a list of the airfoils they published.

Table A-8: Garabedian and Korn airfoil chart

Their airfoils are included in the following references:

Bauer, F., Garabedian, P., and Korn, D., A Theory of Supercritical Wing Sections with Computer Programs and Examples, Lecture Notes in Economics and Mathematical Systems, Vol. 66, Springer-Verlag, 1972.

Bauer, F., Garabedian, P., Jameson, A., and Korn, D., Supercritical Wing Sections II, A Handbook, Lecture Notes in Economics and Mathematical Systems, Vol. 108, Springer-Verlag, 1975.

Bauer, F., Garabedian, P., and Korn, D., Supercritical Wing Sections III, Lecture Notes in Economics and Mathematical Systems, Vol. 150, Springer-Verlag, 1977.

A.2 Classic Bodies of Revolution

Bodies of revolution form the basis for a number of shapes used in aerodynamic design and are also often used in comparing computational methods. The bodies defined in this section are generally associated with supersonic aerodynamics.

A.2.1 Summary of Relations

The body radius r is given as a function of x, r/l = f(x/l). Once r is known, a number of other values characterizing the shape can be determined.

The cross-sectional area and derivatives are

$$S(x)=\pi r^2\text{,}\text{(A-39)}$$

$$\frac{dS}{dx}=2\pi r\frac{dr}{dx}\text{,}\text{(A-40)}$$

and

$$\frac{d^{2}S}{d{x}^2}=2\pi [{\left(\frac{dr}{dx}\right)}^2+r\frac{d^{2}r}{d{x}^2}]\text{.}\text{(A-41)}$$

Basic integrals for volume, surface area, and length along the contour are

$$V={\int }_0^{l}S(x)dx\text{,}\text{(A-42)}$$

$$S_{wet}=2\pi {\int }_0^{l}r(x)dx\text{,}\text{(A-43)}$$

and

$$p(\overline{x})={\int }_0^l\sqrt{1+{\left(\frac{dr}{dx}\right)}^2}dx\text{.}\text{(A-44)}$$

Note that the incremental values can be found by changing the lower limit of the integrals.

The local longitudinal radius of curvature is given by

$$R(x)=\frac{{[1+{\left(\frac{dr}{dx}\right)}^2]}^{3/2}}{\left|\frac{d^{2}r}{d{x}^2}\right|}\text{.}\text{(A-45)}$$

Several simple shapes are also of interest in addition to those presented in detail in the subsections below. These simple shapes are: parabolic spindle, equation (A-46); ellipsoid of revolution, equation (A-47); and the power law body, equation (A-48):

$$\frac{r}{l}=4\frac{r_{\text{mid}}}{l}\frac{x}{l}(1-\frac{x}{l})\text{(A-46)}$$

$$\frac{r}{l}=2\frac{r_{\text{mid}}}{l}\sqrt{\frac{x}{l}(1-\frac{x}{l})}\text{(A-47)}$$

$$\frac{r}{l}=\frac{r_0}{l}{\left(\frac{x}{x_N}\right)}^n\text{(A-48)}$$

In equation (A-48), x_N is the nose length, and r₀ is the radius at x = x_N. The nose is blunt for 0 < n < 1.

Two references that discuss geometry of bodies of revolution are listed below. Krasnov’s reference discusses the spherical nose cap, which is another common shape.

Krasnov, N. F., Aerodynamics of Bodies of Revolution, edited and annotated by D. N. Morris, American Elsevier, New York, 1970.

Johns Hopkins University Applied Physics Laboratory. Handbook of Supersonic Aerodynamics, Vol. 3, Sec. 8, “Bodies of Revolution,” NAVWEPS Report 1488. October 1961.

A.2.2 Tangent/Secant Ogives

The tangent or secant ogives are frequently used shapes in supersonic aerodynamics. The nomenclature is illustrated in figure A-1.

Note that the ogive is actually the arc of a circle, and when δ_r = 0, the ogive ends tangent to the body, meaning that δ_r = 0 represents the tangent ogive body. If δ_r = δ_N, the cone-cylinder is recovered. If δ_r = 0 and δ_N = 90°, the spherical cap case is obtained.

The expression for the radius r is determined using three basic constants for a particular case:

$$A=\frac{r_0}{l}\left(\frac{\cos {\delta }_N}{\cos {\delta }_r-\cos {\delta }_N}\right)\text{(A-49)}$$

$$B=2\frac{r_0}{l}\left(\frac{\sin {\delta }_N}{\cos {\delta }_r-\cos {\delta }_N}\right)\text{(A-50)}$$

$$C=\frac{r_0}{l}\text{(A-51)}$$

The radius is then given by

$$\begin{array}{rlrl}\frac{r}{l}& =\sqrt{A^2+B\left(\frac{x}{l}\right)-{\left(\frac{x}{l}\right)}^2}-A& & 0\text{less than}\frac{x}{l}\text{less than}\frac{x_N}{l}\\ & =C& & \frac{x_N}{l}\text{less than}\frac{x}{l}\text{less than}1\end{array}\text{(A-52)}$$

Next, x_N must be determined. For a tangent ogive (δ_r = 0), the ogive can be defined by specifying either x_N/r₀ or δ_N. The other value can then be found as follows.

Given δ_N,

$$\frac{x_N}{r_0}=\frac{\sin {\delta }_N}{1-\cos {\delta }_N}\text{.}\text{(A-53)}$$

Or given x_N/r₀,

$${\delta }_N={\cos }^{-1}\left[\frac{{\left(\frac{x_N}{r_0}\right)}^2-1}{{\left(\frac{x_N}{r_0}\right)}^2+1}\right]\text{.}\text{(A-54)}$$

For the secant ogive, the simplest analytical procedure is to define the ogive in terms of δ_N and δ_r and then to find x_N/l from

$$\frac{x_N}{l}=\frac{r_0}{l}\left(\frac{\sin {\delta }_N-\sin {\delta }_r}{\cos {\delta }_r-\cos {\delta }_N}\right)\text{.}\text{(A-55)}$$

If x_N/l is not satisfactory, δ_N and δ_r can be adjusted by trial and error to obtain the desired nose length. A program can be set up to handle this process quite simply.

The first and second derivatives are then given by

$$\frac{d(r/l)}{d(x/l)}=\frac{B-2(x/l)}{2[(r/l)+A]}\text{(A-56)}$$

and

$$\frac{d^2(r/l)}{d(x/l{)}^2}=-\frac{[B-2(x/l){]}^2}{4[(r/l)+A{]}^3}-\frac{1}{[(r/l)+A]}\text{.}\text{(A-57)}$$

The relationships between radius and area derivatives given in section A.2.1 are then used to complete the calculation.

A.2.3 The Von Kármán Ogive

This shape produces minimum wave drag for a specified base area and length, according to slender body theory. This ogive has a very slightly blunted nose, and it is described in a book by Ashley and Landahl.^[9]

In this case, it is convenient to work with the cross-sectional area and a new independent variable:

$$\theta ={\cos }^{-1}[2\left(\frac{x}{x_N}\right)-1]\text{(A-58)}$$

or

$$\frac{x}{x_N}=\frac{1}{2}(1+\cos \theta )\text{(A-59)}$$

where the nose is at θ = π and the base is located at θ = 0.

Here we use x_N to denote the “nose length” or length of the ogive and allow this shape to be part of an ogive-cylinder geometry. The shape is then given as

$$\frac{S(x)}{l^2}=\frac{S_B}{l^2}[1-\frac{\theta }{\pi }+\frac{\sin 2\theta }{2\pi }]\text{(A-60)}$$

and

$$\frac{r}{l}=\sqrt{\frac{S/l^2}{\pi }}\text{(A-61)}$$

where S_B is the prescribed base area and l is the total length.

Defining nondimensional area and distance from the nose as:

$$\overline{S}=\frac{S}{l^2},\overline{x}=\frac{x}{l},\text{(A-62)}$$

we have

$$\frac{d\overline{S}}{d\theta }=-\frac{{\overline{S}}_B}{\pi }[1-\cos 2\theta ]\text{,}\text{(A-63)}$$

$$\frac{d^2\overline{S}}{d{\theta }^2}=-\frac{2}{\pi }{\overline{S}}_B\sin 2\theta \text{,}\text{(A-64)}$$

and

$$\begin{array}{c}\frac{d\overline{S}}{d\overline{x}}={\overline{S}}^{\mathrm{\prime }}=\frac{4}{\pi }\left(\frac{l}{x_N}\right){\overline{S}}_B\sin \theta \\ \frac{d^2\overline{S}}{d{\overline{x}}^2}={\overline{S}}^{\mathrm{\prime }\mathrm{\prime }}=-\frac{8}{\pi }{\left(\frac{l}{x_N}\right)}^2\frac{{\overline{S}}_B}{\tan \theta }\end{array}\text{.}\text{(A-65)}$$

The radius derivatives are then computed by

$$\frac{d\overline{r}}{d\overline{x}}=\frac{{\overline{S}}^{\mathrm{\prime }}}{2\pi \overline{r}},\frac{d^2\overline{r}}{d{\overline{x}}^2}=\frac{{\overline{S}}^{\mathrm{\prime }\mathrm{\prime }}}{2\pi \overline{r}}-\frac{{\overline{r}}^{\mathrm{\prime }2}}{\overline{r}}\text{.}\text{(A-66)}$$

A.2.4 The Sears–Haack Body

This is the minimum wave drag shape for a given length and volume according to slender body theory as discussed in Chapter 9. The body is closed at both ends, has a very slightly blunted nose, and is symmetric about the mid-point. It is described by Ashley and Landahl.^[10]

Although the notation used in the previous section A.2.3 for the Von Kármán Ogive could be used, it is more common to describe the Sears-Haack body in the manner presented below. This form uses the fineness ratio, f = l/d_max to scale the shape. However, it is important to realize that the Sears-Haack shape is the minimum drag body for a specified volume and length, not for a specified fineness ratio. The minimum drag body for a specified fineness ratio is described in section A.2.5 below.

In defining

$$\varsigma =1-2\left(\frac{x}{l}\right),\text{(A-67)}$$

the Sears–Haack body is defined as

$$\frac{r}{l}=\frac{1}{2f}{(1-{\varsigma }^2)}^{3/4}\text{.}\text{(A-68)}$$

The derivatives are given by

$$\frac{d(r/l)}{d(x/l)}=\frac{3\varsigma }{1-{\varsigma }^2}\left(\frac{r}{l}\right)\text{(A-69)}$$

and

$$\frac{d^2(r/l)}{d(x/l{)}^2}=-\left(\frac{1}{1-{\varsigma }^2}\right)[\varsigma \frac{d(r/l)}{d(x/l)}+6\left(\frac{r}{l}\right)]\text{.}\text{(A-70)}$$

The fineness ratio is related to the length and volume by

$$f=\sqrt{\frac{3{\pi }^2}{64}\frac{l^3}{V}}.\text{(A-71)}$$

In terms of f and either V or l, the other value can be found based on the following givens.

Given f and l:

$$V=\frac{3{\pi }^2}{64}\frac{l^3}{f^2}\text{(A-72)}$$

Given f and V:

$$l={\left[V\frac{64}{3{\pi }^2}{f}^2\right]}^{1/3}\text{(A-73)}$$

The relationships between radius and area derivatives given in section A.2.1 are then used to complete the calculation.

A.2.5 The Haack–Adams Bodies

The Haack–Adams bodies define a number of minimum drag shapes, as described by M.C. Adams.^[11] These bodies correspond to the following cases:

1. Given length, base area, and contour passing through a specifically located radius
2. Given length, base area, and maximum area
3. Given length, base area, and volume

In case I, the specified radius will not necessarily be the maximum radius.

The notation used in TN-2550 is employed in the equations, leading to the following definitions:

$$S=4\frac{\overline{S}(x)}{l^2},B=4\frac{S_{BASE}}{l^2},A=4\frac{S_A}{l^2},V=8\left(\frac{\overline{V}}{l^3}\right)\text{(A-74)}$$

where S(x) is the area, S_A corresponds to either the specified area at a given location or the maximum area, and V is the volume. The independent variable is defined with its origin at the body midpoint, as seen in equation (A-75).

$$\varsigma =2\left(\frac{x}{l}\right)-1\text{(A-75)}$$

The location of the specified radius (case I) and maximum radius (case II) is designated C and given in ζ coordinates. When referring to the x coordinate, this value is designated C_x.

What follows are the standard forms of equations for each case.

Case I: Given S_BASE, S_A, and C_x

$$\begin{array}{rl}\frac{\pi S}{B}=& [\frac{\pi A}{B}-{\cos }^{-1}(-c)]\frac{\sqrt{1-{\varsigma }^2}(1-c\varsigma )}{{(1-c^2)}^{3/2}}+\frac{\sqrt{1-{\varsigma }^2}(\varsigma -c)}{(1-c^2)}\\ & +[\frac{\pi A}{B}-{\cos }^{-1}(c)-c\sqrt{1-c^2}]\frac{(\varsigma -c{)}^2}{(1-c{)}^2}\ln N+{\cos }^{-1}(-\varsigma )\end{array}\text{(A-76)}$$

where

$$N=\frac{1-c\varsigma -\sqrt{1-c^2}\sqrt{1-{\varsigma }^2}}{|\varsigma -c|}\text{(A-77)}$$

Case II: Given S_BASE and S_MAX

First, find the location of the maximum thickness from the implicit relation.

$$f(c)=0=\frac{\pi A}{R}c-\sqrt{1-c^2}-c{\cos }^{-1}(-c)\text{(A-78)}$$

Use Newton’s iteration

$$c^{i+1}=c^i-\frac{f\left(c^i\right)}{f^{\mathrm{\prime }}\left(c^i\right)}\text{(A-79)}$$

where

$$f^{\mathrm{\prime }}(c)=\frac{\pi A}{B}-{\cos }^{-1}(-c).\text{(A-80)}$$

An initial guess of c = 0 is sufficient to start the iteration. Given c, the relation for the area is

$$\frac{\pi S}{B}=\frac{\sqrt{1-{\varsigma }^2}}{c}+\frac{(\varsigma -c{)}^2}{c\sqrt{1-c^2}}\ln N+{\cos }^{-1}(-\varsigma )\text{(A-81)}$$

where N is the same function as given in case I.

Case III: Given S_BASE and V

$$\frac{\pi S}{B}=\frac{8}{3}[\frac{V}{B}-1]{(1-{\varsigma }^2)}^{3/2}+\varsigma \sqrt{1-{\varsigma }^2}+{\cos }^{-1}(-\varsigma )\text{(A-82)}$$

The maximum thickness for this case is located at

$$e=\frac{1}{4(V/B-1)}\text{,}\text{(A-83)}$$

and in x coordinates, it is located at:

$$e_x=\frac{1}{2}(1+e)\text{.}\text{(A-84)}$$

Note that if S_BASE = 0, the Sears–Haack body is recovered.

A.3 Cross-Section Geometries for Bodies

The axisymmetric bodies described above can be used to define longitudinal lines for aerodynamic bodies. However, many aerodynamic bodies are not axisymmetric because the fuselage cross section is not typically round. In this section, we define a class of cross section shapes that can be used to develop more realistic aerodynamic models. In particular, they have been used to study geometric-shaping effects on forebody aerodynamic characteristics using an analytical forebody model with the ability to produce a wide variation of shapes. This generic model makes use of the equation of a superellipse to define cross-sectional geometry. The superellipse, used previously to control flow expansion around wing leading edges, can recover a circular cross section and can produce elliptical cross sections and chine-shaped cross sections. Thus it can be used to define a variety of different cross-sectional shapes.

The superellipse equation for a cross section is

$${\left(\frac{z}{b}\right)}^{2+n}+{\left(\frac{y}{a}\right)}^{2+m}=1\text{(A-85)}$$

where n and m are adjustable coefficients that control the surface slopes at the top and bottom planes of symmetry and the chine leading edge. The constants a and b correspond to the maximum half-breadth (the maximum width of the body) and the upper or lower centerlines, respectively. Depending on the value of n and m, the equation can be made to produce all the shapes described above. The case n = m = 0 corresponds to the standard ellipse. The body is circular when a = b.

When n = -1, the sidewall is linear at the maximum half-breadth line, forming a distinct crease line. When n < -1, the body cross section takes on a cusped or chine-like shape. As n increases, the cross section starts to become rectangular.

The derivative of z/b with respect to y/a is

$$\frac{d\overline{z}}{dy}=-\frac{\left(\frac{2+m}{2+n}\right)}{{[1-{\overline{y}}^{(2+m)}]}^{\left(\frac{1+n}{2+n}\right)}}\text{(A-86)}$$

where z = z / b and y = y / a. As y → 1, the slope becomes

$$\frac{d\overline{z}}{d\overline{y}}=\{\begin{array}{ll}\mathrm{\infty }& n>-1\\ 0& n<-1\\ -(2+m){\overline{y}}^{1+m}& n=-1\end{array}\text{(A-87)}$$

Figure A-2 shows a quadrant of the cross section for various values of n ranging from a chine to a rectangle.

Different cross sections can be used above and below the maximum half-breadth line. Even more generality can be provided by allowing n and m to be functions of the axial distance x. The parameters a and b can also be functions of the planform shape and can be varied to study planform effects. Notice that when n = -1, the value of m can be used to control the slope of the sidewall at the crease line. Observe that large positive values of n drive the cross-sectional shape to approach a rectangular or square shape.

Connecting various cross-sectional shapes is part of the subject of lofting. A good discussion of lofting can be found in the book by Raymer,^[12] who worked at North American Aviation (actually part of Rockwell), where Liming literally wrote the book^[13] on the analytic definition of aircraft lines. Liming’s book emphasizes conic sections, and the examples in the book are for the P-51 Mustang.

A.4 Planform Analysis

Several local and integral planform properties are of interest in aerodynamic analysis. They are summarized in this section. (Note: Biplanes use the total area of both wings as the reference area.) For a more complete presentation, see DATCOM (https://www.pdas.com/datcom.html).

Figure A-3 illustrates the standard nomenclature. The local values are the leading- and trailing-edge locations, x_LE(y) and x_TE(y), the local chord, c(y), and the leading- and trailing-edge sweep angles: Λ_LE(y) and Λ_TE(y).

Assuming the planform is symmetric, the integral properties are as follows.

(1) Planform Area, S

$$S=2{\int }_0^{b/2}c(y)dy\text{(A-88)}$$

(2) Mean aerodynamic chord (mac)

$$\overline{c}=\frac{2}{S}{\int }_0^{b/2}{c}^2(y)dy\text{(A-89)}$$

(3) X position of centroid of area, x_cen

$$x_{cen}=\frac{2}{S}{\int }_0^{b/2}c(y)\{x_{LE}(y)+\frac{c(y)}{2}\}dy\text{(A-90)}$$

(4) Spanwise position of mac

$$y_{\text{mac}}=\frac{2}{S}{\int }_0^{b/2}yc(y)dy\text{(A-91)}$$

(5) Leading-edge location of mac

$$x_{L{E}_{\text{mac}}}=\frac{2}{S}{\int }_0^{b/2}{x}_{LE}(y)c(y)dy\text{(A-92)}$$

In addition, the following derived quantities are often of interest.

Aspect ratio:

$$AR=\frac{b^2}{S_{ref}}\text{(A-93)}$$

Average chord:

$$c_A=\frac{S_{ref}}{b}\text{(A-94)}$$

Taper ratio:

$$\lambda =\frac{c_T}{c_R}\text{(A-95)}$$

S_ref is usually chosen to be equal to the area of a basic reference trapezoidal planform, and thus the actual planform area, S, may not equal S_ref.

When considering two areas, recall that the centroid of the combined surfaces is

$$\begin{array}{rl}& S\overline{x}=S_1{\overline{x}}_1+S_2{\overline{x}}_2\\ & S\overline{y}=S_1{\overline{y}}_1+S_2{\overline{y}}_2\end{array}\text{.}\text{(A-96)}$$

For a standard trapezoidal wing planform, it is convenient to collect a set of formulas related to its geometric characteristics using a standard nomenclature for geometric parameters shown in figure A-4.

For a wing section at span station y, the leading and trailing edge x coordinates are given by equation (A-97).

$$\begin{array}{rl}& x_{LE}(y)=x_{L{E}_0}+y\tan {\mathrm{\Lambda }}_{LE}(y)\\ & x_{TE}(y)=x_{T{E}_0}+y\tan {\mathrm{\Lambda }}_{TE}(y)\end{array}\text{,}\text{(A-97)}$$

and the local chord is

$$\frac{c(y)}{c_R}=1-(1-\lambda )\eta \text{(A-98)}$$

where

$$y=\frac{b}{2}\eta \text{or}\eta =\frac{y}{b/2}\text{and}\lambda =\frac{c_T}{c_R}\text{.}\text{(A-99)}$$

The sweep at any element line can be found in terms of the sweep at any other by

$$\tan {\mathrm{\Lambda }}_n=\tan {\mathrm{\Lambda }}_m-\frac{4}{AR}[(n-m)\left(\frac{1-\lambda }{1+\lambda }\right)]\text{(A-100)}$$

where n and m are fractions of the local chord. An alternate formula is available using the trailing-edge sweep angle:

$$\tan {\mathrm{\Lambda }}_n=(1-n)\tan {\mathrm{\Lambda }}_{LE}+n\tan {\mathrm{\Lambda }}_{TE}\text{.}\text{(A-101)}$$

The integral and other relations are given by

$$\begin{array}{rl}S& =\frac{b}{2}{c}_R(1+\lambda )\\ c_{\text{ave}}& =\frac{S}{b}\\ \frac{\overline{c}}{c_R}& =\frac{2}{3}\left(\frac{1+\lambda +{\lambda }^2}{1+\lambda }\right)\\ AR& =\frac{b^2}{S}=\frac{b/2}{c_R}\left(\frac{4}{1+\lambda }\right)\\ y_{\text{mac}}& =\frac{b}{6}\left(\frac{1+2\lambda }{1+\lambda }\right)\\ \frac{x_{L{E}_{\text{mac}}}}{c_R}& =\frac{x_{L{E}_0}}{c_R}+\left(\frac{1+2\lambda }{12}\right)AR\tan {\mathrm{\Lambda }}_{LE}\\ x_{cen}& =x_{L{E}_{\text{mac}}}+\frac{\overline{c}}{2}\end{array}\text{.}\text{(A-102)}$$

When computing the projected planform area of an entire configuration, the following formula is useful:

$$S=\sum _{k=1}^{k=N}(y_{k+1}+y_k)(x_{k+1}-x_k)\text{(A-103)}$$

where the standard nomenclature is shown in figure A-5.

At k = N, y_k+1,xk+1 refer to the initial points y₁,x₁. For normal planforms, y_n+1 = y₁ = 0, so that the summation can be terminated at N-1. This formula assumes planform symmetry and provides the total planform area with only one side of the planform used in the computation.

A.4.1 A Note on Reference Chords and Aerodynamic Center Location

Looking at the aerodynamics literature, especially the older literature and textbooks, you find that the nomenclature for reference chords is not uniform. However, there is a standard in use today in the US industry. The issue is what value to use when nondimensionalizing the pitching moment:

$$C_m=\frac{M}{q{S}_{ref}{c}_{ref}}\text{.}\text{(A-104)}$$

The USAF DATCOM defines the standard. They define c_ref to be the mean aerodynamic chord or mac:

$$\overline{c}=\frac{2}{S}{\int }_0^{b/2}{c}^2(y)dy\text{(A-105)}$$

which, for a straight tapered wing, is

$$\frac{\overline{c}}{c_r}=\frac{2}{3}\left(\frac{1+\lambda +{\lambda }^2}{1+\lambda }\right)\text{.}\text{(A-106)}$$

The use of the symbol  [latex]\bar{c}[/latex]  to represent the mean aerodynamic chord is confusing because  [latex]\bar{c}[/latex]  is sometimes used to represent the average chord, c_a = S/b, in other literature. For example, ESDU Item 76003 uses  [latex]\bar{c}[/latex]  to be the standard (or geometric) mean chord, which I have defined here as the average chord. They use c (with two overlines) as the mac defined by equation (A-89). The main problem occurs when the report or paper does not specifically define the terms used. For example, Lamar and Alford^[14] use equation (A-89) to nondimensionalize the pitching moment, but they call it the “mean geometric chord.”

Another issue to consider is that the mac does not necessarily lie on the actual planform if the wing is not a simple straight tapered configuration.

A.5 Conical Camber

An important class of camber distributions is associated with the planform and not the airfoil. Conical camber has been widely used, and many forms can be found in the literature. However, the NACA defined a specific type of conical camber that is known as NACA conical camber. The most recent example of NACA conical camber is the F-15 wing. It improves the drag characteristics of wings in the subsonic and transonic flow region, even though it was developed to reduce the drag at supersonic speeds!

The key references on this topic are listed below.

Hall, C. F., “Lift, Drag, and Pitching Moment of Low Aspect Ratio Wings at Subsonic and Supersonic Speeds,” NACA RM-A53A30, 1953. https://ntrs.nasa.gov/citations/19930086624

This report provided the original mathematical definition of NACA conical camber. It also provided a large range of test conditions for which the camber was effective.

Boyd, J. W., Migotsky, E., and Wetzel, B. E., “A Study of Conical Camber for Triangular and Swept Back Wings,” NASA RM-A55G19, Nov. 1955. https://ntrs.nasa.gov/citations/19930090334

This report provided more details of the derivation of the formulas for NACA conical camber, and it corrected errors in the equations presented in the first report. Additional experimental results were also presented.

An advanced form of conical camber addressing cases of so-called supercritical crossflows is supercritical conical camber (SC3). It is described in a report by Mason^[15] and in Mason’s AIAA paper.^[16]

A.6 Three-Dimensional Wing Geometry

Wing geometry is often defined by interpolating between airfoil sections specified at particular spanwise stations. Some care should be taken to interpolate properly. Program WNGLFT provides an example of a lofting scheme to provide wing ordinates at any desired location. It can be used to provide wing ordinates for a wide class of wings. In fact, it will produce a very good approximation of the wing design employed by a successful Navy airplane.