Geometry API

An abstract type to define custom spacing distributions.

Foil(x, y, name = "")
Foil(coordinates, name = "")

Structure consisting of foil coordinates in 2 dimensions with an optional name.

The coordinates should be provided in counter-clockwise format, viz. from the trailing edge of the upper surface to the trailing edge of the lower surface.

HyperEllipseFuselage(;

Define a fuselage based on the following hyperelliptical-cylindrical parameterization.

Nose: Hyperellipse $z(ξ) = (1 - ξ^a)^{(1/a)}$

Cabin: Cylindrical $z(ξ) = 1$

Rear: Hyperellipse $z(ξ) = (1 - ξ^b)^{(1/b)}$

Arguments

• radius :: Real = 1.: Fuselage radius (m)
• length :: Real = 6.: Fuselage length (m)
• x_a :: Real = 1.: Location of front of cabin as a ratio of fuselage length ∈ [0,1]
• x_b :: Real = 0.: Location of rear of cabin as a ratio of fuselage length ∈ [0,1]
• c_nose :: Real = 0.: Curvature of nose in terms of hyperellipse parameter $a$
• c_rear :: Real = 1.: Curvature of rear in terms of hyperellipse parameter $b$
• d_nose :: Real = 1.: "Droop" or "rise" of nose from front of cabin centerline (m)
• d_rear :: Real = 1.: "Droop" or "rise" of rear from rear of cabin centerline (m)
• position :: Vector{Real} = zeros(3): Position (m)
• angle :: Real = 0.: Angle of rotation (degrees)
• axis :: Vector{Real} = [0, 1 ,0]: Axis of rotation, y-axis by default
• affine :: AffineMap = AffineMap(AngleAxis(deg2rad(angle), axis...), position): Affine mapping for the position and orientation via CoordinateTransformations.jl (overrides angle and axis if specified)

Wing(
    chords, 
    foils :: Vector{Foil}, 
    twists, 
    spans, 
    dihedrals, 
    sweeps,
    symmetry = false,
    flip     = false,
    position = zeros(3),
    angle    = 0.,
    axis     = [0.,1.,0.],
)

Definition for a Wing consisting of $N+1$ Foils, their associated chord lengths $c$ and twist angles $ι$, for $N$ sections with span lengths $b$, dihedrals $δ$ and leading-edge sweep angles $Λ_{LE}$, with all angles in degrees. Optionally, specify translation and a rotation in angle-axis representation for defining coordinates in a global axis system. Additionally, specify Boolean arguments for symmetry or reflecting in the $x$-$z$ plane.

Arguments

• chords :: Vector{Real}: Chord lengths (m)
• foils :: Vector{Foil} = fill(naca4(0,0,1,2), length(chords)): Foil shapes, default is NACA 0012.
• spans :: Vector{Real} = ones(length(chords) - 1) / (length(chords) - 1): Span lengths (m), default yields total span length 1.
• dihedrals :: Vector{Real} = zero(spans): Dihedral angles (deg), default is zero.
• sweeps :: Vector{Real} = zero(spans): Sweep angles (deg), default is zero.
• w_sweep :: Real = 0.: Chord ratio for sweep angle e.g., 0 = Leading-edge sweep, 1 = Trailing-edge sweep, 0.25 = Quarter-chord sweep
• symmetry :: Bool = false: Symmetric in the $x$-$z$ plane
• flip :: Bool = false: Flip coordinates in the $x$-$z$ plane
• position :: Vector{Real} = zeros(3): Position (m)
• angle :: Real = 0.: Angle of rotation (degrees)
• axis :: Vector{Real} = [0.,1.,0.]: Axis of rotation
• affine :: AffineMap = AffineMap(AngleAxis(deg2rad(angle), axis...), position): Affine mapping for the position and orientation via CoordinateTransformations.jl (overrides angle and axis if specified)

WingMesh(
    wing :: AbstractWing, 
    n_span :: Vector{Integer}, n_chord :: Integer;
    span_spacing :: AbstractSpacing = symmetric_spacing(wing)
)

Define a container to generate meshes and panels for a given Wing with a specified distribution of number of spanwise panels, and a number of chordwise panels.

Optionally a combination of AbstractSpacing types (Sine(), Cosine(), Uniform()) can be provided to the named argument span_spacing, either as a singleton or as a vector with length equal to the number of spanwise sections. By default, the combination is [Sine(), Cosine(), ..., Cosine()].

For surface coordinates, the wing mesh will have (nchord - 1) * 2 chordwise panels from TE-LE-TE and (nspan * 2) spanwise panels.

WingSection(; 
    area, aspect, taper
    dihedral, sweep, w_sweep,
    root_twist, tip_twist,
    position, angle, axis,
    symmetry, flip,
    root_foil, tip_foil,
    root_control, tip_control,
)

Define a Wing in the $x$-$z$ plane, with optional Boolean arguments for symmetry and flipping in the plane.

Arguments

• area :: Real = 1.: Area (m²)
• aspect :: Real = 6.: Aspect ratio
• taper :: Real = 1.: Taper ratio of tip to root chord
• dihedral :: Real = 1.: Dihedral angle (degrees)
• sweep :: Real = 0.: Sweep angle (degrees)
• w_sweep :: Real = 0.: Chord ratio for sweep angle e.g., 0 = Leading-edge sweep, 1 = Trailing-edge sweep, 0.25 = Quarter-chord sweep
• root_twist :: Real = 0.: Twist angle at root (degrees)
• tip_twist :: Real = 0.: Twist angle at tip (degrees)
• root_foil :: Foil = naca4((0,0,1,2)): Foil at root
• tip_foil :: Foil = root_foil: Foil at tip. Defaults to root foil.
• root_control :: NTuple{2} = (0., 0.75): (Angle, hinge ratio) for adding a control surface at the root.
• tip_control :: NTuple{2} = root_control: (Angle, hinge ratio) for adding a control surface at the tip. Defaults to root control's settings.
• symmetry :: Bool = false: Symmetric in the $x$-$z$ plane
• flip :: Bool = false: Flip coordinates in the $x$-$z$ plane
• position :: Vector{Real} = zeros(3): Position (m)
• angle :: Real = 0.: Angle of rotation (degrees)
• axis :: Vector{Real} = [0.,1.,0.]: Axis of rotation
• affine :: AffineMap = AffineMap(AngleAxis(deg2rad(angle), axis...), position): Affine mapping for the position and orientation via CoordinateTransformations.jl (overrides angle and axis if specified)

affine(
    foil :: Foil, 
    angle, vector
)

Perform an affine transformation on the coordinates of a Foil by a 2-dimensional vector $\mathbf v$ and angle $θ$.

arc_length(foil :: Foil)

Compute the arc-length of a Foil.

aspect_ratio(wing :: AbstractWing)

Compute the aspect ratio of an AbstractWing.

camber(foil :: Foil, x_by_c)

Obtain the camber value of a Foil at a specified $(x/c)$.

camber_CST(α_c, α_t,
           Δz :: Real,
           n :: Integer = 40)

Define a cosine-spaced foil with $2n$ points using the Class Shape Transformation method on a Bernstein polynomial basis for the camber and thickness coordinates.

The foil is defined by arrays of coefficients $(α_c,~ α_t)$ for the upper and lower surfaces, trailing-edge spacing values $(Δz_u,~Δz_l)$, and a coefficient for the leading edge modifications at the nose.

camber_coordinates(wing :: WingMesh, n_span = wing.num_span, n_chord = wing.num_chord)

Generate the camber coordinates of a WingMesh with default spanwise $n_s$ and chordwise $n_c$ panel distributions from the mesh.

camber_line(foil :: Foil, n :: Integer = 40)

Get the camber line of a Foil. Optionally specify the number of points for linear interpolation, default is 40.

camber_panels(wing_mesh :: WingMesh)

Generate the camber mesh as a matrix of Panel3D from a WingMesh.

camber_thickness(foil :: Foil, num :: Integer)

Compute the camber-thickness distribution of a Foil with cosine interpolation. Optionally specify the number of points for interpolation, default is 40.

camber_thickness(wing :: Wing, num :: Integer)

Compute the camber-thickness distribution at each spanwise intersection of a Wing. A num must be specified to interpolate the internal Foil coordinates, which affects the accuracy of $(t/c)ₘₐₓ$ accordingly.

camber_thickness_to_CST(coords, num_dvs)

Convert camber-thickness coordinates to a specified number of Bernstein polynomial coefficients under a Class Shape Transformation by performing a least-squares solution.

chord_coordinates(wing :: WingMesh, n_span = wing.num_span, n_chord = wing.num_chord)

Generate the chord coordinates of a WingMesh with default spanwise $n_s$ and chordwise $n_c$ panel distributions from the mesh.

chord_panels(wing_mesh :: WingMesh)

Generate the chord mesh as a matrix of Panel3D from a WingMesh.

control_surface(foil :: Foil, δ, xc_hinge)
control_surface(foil :: Foil; angle, hinge)

Modify a Foil to mimic a control surface by specifying a deflection angle $δ$(in degrees, clockwise-positive convention) and a normalized hingex-coordinate∈ [0,1]in terms of the chord length. A constructor with named argumentsangle, hinge` is provided for convenience.

wetted_area(fuse :: HyperEllipseFuselage, t)

Get the coordinates of a HyperEllipseFuselage given the parameter distribution $t$. Note that the distribution must have endpoints 0 and 1.

coordinates(foil :: Foil)

Generate the array of Foil coordinates.

coordinates_to_CST(coords, num_dvs)

Convert coordinates to a specified number of Bernstein polynomial coefficients under a Class Shape Transformation by performing a least-squares solution.

cosine_interpolation(foil :: Foil, n :: Integer = 40)

Interpolate a Foil profile's coordinates to a cosine by projecting the x-coordinates of a circle onto the geometry with $2n$ points.

interpolate(foil :: Foil, xs)

Linearly interpolate the coordinates of a Foil to a given $x ∈ [0,1]$ distribution.

kulfan_CST(alpha_u, alpha_l,
           (Δz_u, Δz_l) = (0., 0.),
           (LE_u, LE_l) = (0., 0.),
           n            = 40)

Define a cosine-spaced foil with $2n$ points using the Class Shape Transformation method on a Bernstein polynomial basis for the upper and lower coordinates.

The foil is defined by arrays of coefficients $(α_u,~ α_l)$ for the upper and lower surfaces (not necessarily of the same lengths), trailing-edge displacement values $(Δz_u,~ Δz_l)$, and coefficients for leading edge modifications on the upper and lower surfaces at the nose.

leading_edge(wing :: Wing)

Compute the leading edge coordinates of a Wing.

leading_edge_index(foil :: Foil)

Get the index of the leading edge of a Foil. This will be the index of the point with the minimum $x$-coordinate.

lower_surface(foil :: Foil)

Get the lower surface coordinates of a Foil from leading to trailing edge.

maximum_thickness_to_chord(foil :: Foil, num :: Integer)

Compute the maximum thickness-to-chord ratio $(t/c)ₘₐₓ$ and its location $(x/c)$ of a Foil. Returned as the pair $(x/c, (t/c)ₘₐₓ)$.

A num must be specified to interpolate the Foil coordinates, which affects the accuracy of $(t/c)ₘₐₓ$ accordingly, default is 40.

maximum_thickness_to_chord(wing :: Wing, num :: Integer)

Compute the maximum thickness-to-chord ratios $(t/c)ₘₐₓ$ and their locations $(x/c)$ at each spanwise intersection of a Wing.

Returns an array of pairs $[(x/c),(t/c)ₘₐₓ]$, in which the first entry of each pair is the location (normalized to the local chord length at the spanwise intersection) and the corresponding maximum thickness-to-chord ratio at the intersection.

A num must be specified to interpolate the internal Foil coordinates, which affects the accuracy of $(t/c)ₘₐₓ$ accordingly.

mean_aerodynamic_center(wing :: Wing, 
    factor = 0.25; 
    symmetry = wing.symmetry, 
    flip = wing.flip
)

Compute the mean aerodynamic center of a Wing. By default, the factor is assumed to be at 25% from the leading edge, which can be adjusted. Similarly, options are provided to account for symmetry or to flip the location in the $x$-$z$ plane.

mean_aerodynamic_chord(wing :: Wing)

Compute the mean aerodynamic chord of a Wing.

naca4(digits :: NTuple{4, <: Real}, n :: Integer = 40; sharp_TE :: Bool)
naca4(a, b, c, d, n :: Integer = 40; sharp_TE :: Bool)

Generate a Foil of a NACA 4-digit series profile with the specified digits, number of points (40 by default), and a named option to specify a sharp or blunt trailing edge.

Refer to the formula for the digits here: http://airfoiltools.com/airfoil/naca4digit

naca4_coordinates(digits :: NTuple{4, <: Real}, n :: Integer, sharp_TE :: Bool)

Generate the coordinates of a NACA 4-digit series profile with a specified number of points, and a Boolean flag to specify a sharp or blunt trailing edge.

projected_area(wing :: Wing)

Compute the projected area (onto the spanwise plane) of a Wing.

properties(wing :: AbstractWing)

Compute the generic properties of interest (span, area, etc.) of an AbstractWing.

read_foil(
    path :: String; 
    header = true; 
    name = ""
)

Generate a Foil from a file consisting of 2D coordinates with named arguments to skip the header (first line of the file) or assign a name.

By default, the header is assumed to exist and should contain the airfoil name, which is assigned to the name of the Foil.

reflect(foil :: Foil)

Reflect the $y$-coordinates of a Foil about the $y = 0$ line.

rotate(foil :: Foil; 
    angle, 
    center = zeros(2)
)

Rotate the coordinates of a Foil about a 2-dimensional point (default is origin) by the angle $θ$ (in degrees).

scale(foil :: Foil, scale)

Scale the coordinates of a Foil to a scaling value.

span(wing :: Wing)

Compute the planform span of a Wing.

split_surface(foil :: Foil)

Split the Foil coordinates into upper and lower surfaces.

surface_coordinates(wing :: WingMesh, n_span = wing.num_span, n_chord = wing.num_chord)

Generate the surface coordinates of a WingMesh with default spanwise $n_s$ and chordwise $n_c$ panel distributions from the mesh.

surface_panels(
    wing_mesh :: WingMesh, 
    n_s = wing_mesh.num_span, 
    n_c = length(first(foils(wing_mesh.surface))).x
)

Generate the surface panel distribution from a WingMesh with the default spanwise $n_s$ panel distribution from the mesh and the chordwise panel $n_c$ distribution from the number of points defining the root airfoil.

In case of strange results, provide a higher number of chordwise panels to represent the airfoils more accurately

sweeps(wing :: Wing, w = 0.)

Obtain the sweep angles (in radians) at the corresponding normalized chord length ratio $w ∈ [0,1]$.

taper_ratio(wing :: Wing)

Compute the taper ratio of a Wing, defined as the tip chord length divided by the root chord length, independent of the number of sections.

thickness_line(foil :: Foil, n :: Integer = 40)

Get the thickness line of a Foil. Optionally specify the number of points for linear interpolation, default is 40.

trailing_edge(wing :: Wing)

Compute the trailing edge coordinates of a Wing.

translate(foil :: Foil, vector)

Translate the coordinates of a Foil by a 2-dimensional vector $\mathbf v$.

upper_surface(foil :: Foil)

Get the upper surface coordinates of a Foil from leading to trailing edge.

volume(fuse :: HyperEllipseFuselage, ts)

Compute the volume of a HyperEllipseFuselage given the parameter distribution $t$. Note that the distribution must have endpoints 0 and 1.

wetted_area_ratio(
    wing_mesh :: WingMesh, 
    n_s = wing_mesh.num_span, 
    n_c = length(first(foils(wing_mesh.surface))).x
)

Determine the wetted area ratio $S_{wet}/S$ of a WingMesh by calculating the ratio of the total area of the surface panels to the projected area of the Wing.

The wetted area ratio should be slightly above 2 for thin airfoils.

make_panels(foil :: Foil)
make_panels(foil :: Foil, n :: Integer)

Generate a vector of Panel2Ds from a Foil, additionally with cosine interpolation using (approximately) $n$ points if provided.

wetted_area(
    wing_mesh :: WingMesh, 
    n_s = wing_mesh.num_span, 
    n_c = length(first(foils(wing_mesh.surface))).x
)

Determine the wetted area $S_{wet}$ of a WingMesh by calculating the total area of the surface panels.

wetted_area(fuse :: HyperEllipseFuselage, t)

Compute the wetted area of a HyperEllipseFuselage given the parameter distribution $t$. Note that the distribution must have endpoints 0 and 1.

Panel3D(p1, p2, p3, p4)

Four Cartesian coordinates p1, p2, p3, p4 representing corners of a panel in 3 dimensions. The following commutative diagram (math joke) depicts the order:

z → y
↓
x
        p1 —→— p4
        |       |
        ↓       ↓
        |       |
        p2 —→— p3

get_transformation(panel :: AbstractPanel3D)

Generate the mapping to transform a point from global coordinates to an AbstractPanel3D's local coordinate system.

make_panels(xyzs)

Convert an array of coordinates corresponding to a wing, ordered from root to tip and leading-edge to trailing-edge, into panels.

midpoint(panel :: AbstractPanel3D)

Compute the midpoint of an AbstractPanel3D.

normal_vector(panel :: Panel3D)

Compute the normal vector of an AbstractPanel3D.

panel_area(panel :: AbstractPanel3D)

Compute the area of a planar quadrilateral 3D panel.

panel_coordinates(panel :: Panel3D)

Compute the coordinates of a Panel3D.

reflect_xz(panel :: Panel3D)

Reflect a Panel3D with respect to the $x$-$z$ plane of its reference coordinate system.

transform(panel :: Panel3D, rotation, translation)

Perform an affine transformation on the coordinates of a Panel3D given a rotation matrix and translation vector.

transform_normal(panel :: Panel3D, h_l, g_l)

Transform the normal vector $n̂₀$ of a Panel3D about the hinge axis $ĥₗ$ by the control gain $gₗ$.

The transformation is the following: $n̂ₗ = gₗ ĥₗ × n̂₀$ (Flight Vehicle Aerodynamics, M. Drela, Eq. 6.36).

wake_panel(panels :: DenseArray{<: AbstractPanel3D}, bound, U)

Calculate required transformation from the global coordinate system to an to an AbstractPanel3D's local coordinate system.

wetted_area(panels :: Array{Panel3D})

Compute the total wetted area by summing the areas of an array of Panel3D.