Airfoils

Contents

An airfoil is a cross section curve defined by a chord and a thickness, from a family of shapes: the NACA series, simple supersonic shapes, a few analytic families, or points read from a file. Wings and propellers are built from them, and Fuselages, Stacks and Bodies of Revolution can use them too, from the same Choose Type list as the body shapes on Cross Section Curves. The Modify tab of the Wing, Propeller and Body of Revolution then reshapes an airfoil's edges: it closes, trims, caps and flaps them, and shifts, turns and scales the whole section.

How an Airfoil Is Laid Out

An airfoil's curve starts at the trailing edge, runs forward along the lower surface to the leading edge, which it reaches halfway round, and returns along the upper surface to the trailing edge (Figure 1). The chord runs from the leading edge to the trailing edge. On a Wing the airfoil lies in the wing's $XZ$ plane with the chord along $+X$; on a Fuselage or Stack the chord takes the place of the section's width, along $Y$, with the leading edge toward $-Y$.

Figure 1. NACA 2412 on a wing: where its curve starts, which way it runs, and its chord.

Controls Every Airfoil Has

On a Wing the chord comes from the wing's planform, and on a Propeller the chord, thickness and design lift coefficient come from the blade curves; see Wing and Propeller. Between two sections of a propeller blade, OpenVSP builds a true airfoil at every radius when both are of the same family defined by equations, and otherwise blends the two shapes; the Propeller page's Cross Sections explains this, and why to use as few families as possible.

Fit CST fits the airfoil as its family defines it, before any Modify operation. The fit comes with the shift, rotation and scale that put the fitted shape where the original was, and these are combined with the Modify tab's Delta X/C, Delta Y/C, Theta, Scale and Shift LE, so the section keeps its shape and place. If the airfoil has a blunt trailing edge, the fit also sets a SKEW BOTH trailing edge closure of that thickness. The other Modify settings are kept.

Design Lift Coefficient

The design lift coefficient, or ideal lift coefficient $C_{l_i}$, is the lift coefficient at which the flow meets the leading edge smoothly; it is set by the camber. The NACA 5-digit, 5-digit modified, 16-series and 6-series families take it directly, as Ideal CL. The NACA 4-digit and 4-digit modified families are defined by their maximum camber instead, so for them Camber and Ideal CL are alternatives: pick one with the toggle beside it, and the other is computed. The relation is thin airfoil theory's

$$C_{l_i} = \pi A_1, \qquad A_1 = \frac{2}{\pi}\int_0^\pi \frac{dz_c}{dx}\,\cos\theta\,d\theta, \qquad x = \frac{1 - \cos\theta}{2}$$

where $z_c(x)$ is the mean line. For the 4-digit mean line $C_{l_i}$ is proportional to the maximum camber $m$, with a factor that depends on the camber position $p$: at $p = 0.4$, $C_{l_i} = 12.8\,m$. This is how a propeller's CLi curve sets the camber of these families.

The other families have no design lift coefficient. Biconvex airfoils have no camber, and wedge, Karman-Trefftz, CST and file airfoils keep the camber their own parameters or points give them.

The Airfoil Families

Figure 2 shows one of each family defined by equations.

Figure 2. One airfoil of each family defined by equations, all at one scale.

NACA 4-Digit

FOUR_SERIES. The NACA 4-digit airfoil: Camber $m$ (the maximum camber, as a fraction of the chord), CamberLoc $p$ (its position along the chord) and T/C. Name shows the designation, NACA 2412 for $m = 0.02$, $p = 0.4$, T/C 0.12.

NACA 4-Digit Modified

FOUR_DIGIT_MOD. The 4-digit airfoil with two more parameters for its thickness (Figure 3). T/CLoc places the maximum thickness, from 0.2 to 0.6 of the chord; the plain 4-digit airfoil has it at 0.3. LERadIndx sets the leading edge radius: 6 is the 4-digit airfoil's radius, 0 is a sharp leading edge, and the radius grows with the square of the index up to 8, then to three times the normal radius at 9. The designation adds the two as a suffix: NACA 2412-63 has index 6 and maximum thickness at 0.3.

Figure 3. A 12% thick 4-digit modified airfoil with three leading edge radius indices (top) and three positions of maximum thickness (bottom).

NACA 5-Digit and 5-Digit Modified

FIVE_DIGIT and FIVE_DIGIT_MOD. The 4-digit thickness with the 5-digit (non-reflexed) mean line, set by Ideal CL and CamberLoc, the position of maximum camber. NACA 23012 has Ideal CL 0.3 and CamberLoc 0.15. The modified form adds T/CLoc and LERadIndx, as for the 4-digit modified airfoil.

NACA 16-Series

16_SERIES. The NACA 16 thickness form, thickest at half the chord, with the $a = 1$ mean line, which loads the chord uniformly. It is set by Ideal CL and T/C: NACA 16-212 has Ideal CL 0.2 and T/C 0.12. Propellers use it widely.

NACA 6-Series

SIX_SERIES. The NACA 6-series airfoils, computed by the NACA 6-series generation method. Series chooses the thickness form, 63- to 67- or the A forms 63A to 65A; Ideal CL and a set the mean line, where a is the fraction of the chord over which the loading is uniform. The A forms always use $a = 0.8$. T/C may be up to 0.5. A script sets Series with SIX_SERIES_63 ... SIX_SERIES_65A.

Biconvex

BICONVEX. Two parabolic arcs, upper and lower, meeting in sharp edges: $z = \pm 2\,(t/c)\,x(1 - x)$. Only T/C shapes it. It is a common supersonic section.

Wedge

WEDGE. A section of straight lines: a sharp leading and trailing edge, and on each surface a corner at the maximum thickness, or a flat between two corners (Figure 4).

The U parameters do not change the shape. They decide which part of the curve meets which part of the neighboring sections in the loft, so give neighboring wedge sections the same values to make their corners line up.

Figure 4. A wedge with T/C 0.12, Thick X 0.3, Flat Up 0.3, Thick X Low 0.4, Flat Low 0.2 and Camber 0.02. The dots are the corners.

Airfoil File

AF_FILE. Read File reads an airfoil's coordinates in one of three formats:

• Selig (.dat): a name line, then $x$ $z$ pairs from the upper trailing edge forward around the leading edge and back along the lower surface.
• Lednicer (.dat): a name line, a line with the number of upper and lower points, then the upper surface and the lower surface, each from the leading edge.
• OpenVSP (.af): a first line containing AIRFOIL FILE, a name line, a symmetry flag, the number of points on the upper surface (and on the lower, unless symmetric), then the upper points, one line that is skipped (left blank), and the lower points (none for a symmetric airfoil).

OpenVSP splits the airfoil into a mean line and a thickness distribution. Base T/C is the file's thickness, and T/C scales the thickness about the mean line, so changing it keeps the camber.

CST

CST_AIRFOIL. A Class-Shape Transformation airfoil (Kulfan): each surface is

$$z(x) = \sqrt{x}\,(1 - x)\,\sum_{i=0}^{n} A_i\,\binom{n}{i}\,x^i(1 - x)^{n-i}$$

with its own Degree $n$ and coefficients $A_i$, listed under Upper Surface and Lower Surface. The first coefficient sets the leading edge radius, $r = A_0^2/2$, and the last the angle at the trailing edge. Enforce Continuous LE Radius makes the lower surface's first coefficient the negative of the upper's, so the two surfaces share one leading edge radius. Promote raises a surface's degree without changing its shape, which gives more coefficients to work with; Demote lowers it and approximates the shape. The trailing edge is sharp; use the Modify tab to thicken it. T/C is an output.

CST airfoils are smooth and have few parameters, which suits them to optimization (Figure 5). Fit CST on the other families converts an airfoil to one.

Figure 5. NACA 2412 as a CST airfoil of degree 5 on each surface, on the Airfoil tab of a Wing. Each list of coefficients scrolls; Al_0 is greyed because Enforce Continuous LE Radius sets it from Au_0, and T/C is shown, not set.

Karman-Trefftz

KARMAN_TREFFTZ. The conformal map of a circle, from potential flow theory (Figure 6). Epsilon moves the circle's center to thicken the airfoil, Kappa moves it up to camber it, and Tau is the trailing edge angle in degrees; 0 gives a cusp. The chord is scaled to Chord, and T/C is an output. The pressure distribution of these airfoils is known exactly, which makes them useful test cases; the API's GetVKTAirfoilCpDist gives it.

Figure 6. Karman-Trefftz airfoils: Epsilon, Kappa and Tau each varied from the defaults, Epsilon 0.1, Kappa 0.1 and Tau 10.

The Modify Tab

The Wing, Propeller and Body of Revolution have a Modify tab that reshapes the active airfoil. Its operations are applied in a fixed order, each to the result of the one before:

1. Trailing edge closure
2. Trailing edge trim
3. Trailing edge flap
4. Leading edge closure
5. Leading edge trim
6. Trailing edge cap
7. Leading edge cap
8. Rotate and scale, then shift

So a cap rounds the face that a closure or trim made, a flap turns the trimmed trailing edge, and shifting, rotating and scaling move the finished section. The order does not change; to trim and then close, for example, is not possible.

Lengths are given either absolutely, Abs, in model units, or relative to the chord, Rel. The one not chosen is shown, updated to match. Use Rel on a wing whose sections differ in chord, so that a modification keeps its proportion along the span.

Shift, Rotate and Scale

Theta turns the section about its leading edge, positive moving the trailing edge up, toward $+Z$ on a wing. Scale scales it about its leading edge. Delta X/C and Delta Y/C then move it along and across the chord, in chords (Figure 7).

Figure 7. NACA 2412 turned by Theta, moved by Delta X/C and Delta Y/C, and scaled; dashed, before. The dot is the leading edge before.

Shift LE changes where the curve's leading edge falls, not the shape: the point halfway round the curve moves onto the upper surface for positive values and the lower surface for negative ones (Figure 8). The surface's leading edge line, where the wing's upper and lower surfaces meet in its parameters, moves with it. It is for airfoils whose geometric leading edge is not where their curve puts it, such as a drooped or turned airfoil.

Figure 8. The point halfway round NACA 2412's curve, with three values of Shift LE.

Closure

A closure shears one or both surfaces so that an edge becomes a flat face of thickness T: it opens a sharp edge, or, with T 0, closes a blunt one.

At the trailing edge (Figure 9), SKEW LOWER shears the lower surface, SKEW UPPER the upper, and SKEW BOTH each by half; each shears the surface across the chord, so the change grows linearly from nothing at the leading edge. EXTRAPOLATE instead extends both surfaces along their directions at the trailing edge until they are T apart; it lengthens the airfoil without changing anything ahead of the old trailing edge, and acts only where the two surfaces converge. It is the way to close a blunt trailing edge to a point.

Figure 9. Trailing edge closures on NACA 2412, aft part; dashed, before. EXTRAPOLATE is shown on the same airfoil already cut off where it is 0.03 thick, closed to T 0.

At the leading edge (Figure 10) the shear grows from nothing at the trailing edge instead, and opens the leading edge into a flat face T high.

Figure 10. Leading edge closures on NACA 2412, forward part; dashed, before.

Trim

A trim cuts an edge off, leaving a flat face (Figure 11). X cuts at the points that distance from the edge, THICK where the airfoil is T thick. The chord parameter does not change, so a trimmed airfoil is shorter than its chord.

Figure 11. Trailing and leading edge trims on NACA 2412; dashed, before.

Plain Flap

PLAIN deflects the aft part of the airfoil as a plain flap (Figure 12). The hinge is X from the trailing edge (the flap's chord, Abs or Rel) and Y/T of the way across the airfoil there, from the lower surface at 0 to the upper at 1. Deflection is in degrees, positive trailing edge down. The side toward which the flap turns gets a sharp corner, and the other side an arc about the hinge, so the surface stays closed.

The flap works on a Wing, a Propeller and a Body of Revolution.

Figure 12. A plain flap of X/C 0.25 and Y/T 0.5 on NACA 2412, deflected 20 degrees each way; dashed, undeflected.

Caps

A cap shapes the flat face of a blunt edge, whether it comes from the airfoil itself, a closure or a trim (Figure 13). FLAT leaves the face flat. ROUND replaces it with an arc tangent to both surfaces. EDGE replaces it with two straight lines that meet in a point, and SHARP with two curves that meet in a point.

Figure 13. Trailing edge caps on NACA 2412 cut off where it is 0.04 thick; dashed, the flat face.

Three parameters adjust a cap (Figure 14):

• Length stretches it along the chord: 1 is a circular arc for ROUND, and for EDGE and SHARP puts the point half the face's height beyond it.
• Offset moves the tip across the face, as a fraction of its height. Positive is toward the upper surface at the trailing edge and toward the lower surface at the leading edge.
• Strength, for SHARP only, sets how full the two curves are: 0 makes them straight, like EDGE, and 1 fullest.

Figure 14. ROUND with three Lengths and three Offsets, and SHARP at Length 2 with three Strengths, on the same trailing edge.

Scripting

The API reaches an airfoil as it does any cross section (see Cross Section Curves), and has functions of its own for airfoils:

The Modify tab's parameters are found by name with GetXSecParm: Theta, Scale, DeltaX, DeltaY, ShiftLE, and TE_Close_Type, TE_Trim_Type, TE_Cap_Type and their LE_ counterparts, and TE_Flap_Type, with the values in the XSEC_CLOSE_TYPE, XSEC_TRIM_TYPE, XSEC_FLAP_TYPE and end cap enumerations. The Abs/Rel toggles are TE_Close_AbsRel, TE_Trim_AbsRel and TE_Flap_AbsRel (with ABS or REL), and the values they choose between are:

The flap's Y/T and Deflection are TE_Flap_Y_Frc and TE_Flap_Deflection (Flap), and a cap's Length, Offset and Strength are TE_Cap_Length, TE_Cap_Offset and TE_Cap_Strength (Cap). The leading edge has the same names with LE_ in place of TE_, except for the flap.