Vortex Lattice Analysis: Method

Vortex Lattice Analysis: the Lattice, Axes and Signs, the Fuselage Model, What Was Verified and the Limits

The tool finds the air loads on a whole aircraft with the vortex-lattice method: the lifting surfaces are covered with a lattice of horseshoe vortices whose strengths make the flow tangent to the surfaces. It is the method of the NASA vortex-lattice programs of the 1970s and of the program AVL, which is the reference for what the tool can do. The solver was written for this site from the published theory; no part of another program's source code was used. What the tool reads and writes of AVL is its documented geometry-file format.

The theory is linear and inviscid. It gives lift, induced drag, the distribution of load along the span, moments and all stability and control derivatives well for attached flow at small angles. It does not predict stall, profile drag or separated flow: those enter only through the section values you enter (c_l,max and a strip drag polar). Use it for preliminary design and for teaching, not as a substitute for a wind-tunnel test or a certification analysis.

Getting a result quickly

• Simple mode describes the aircraft with a few numbers per surface: span, root chord, taper ratio, sweep, dihedral, incidence, twist and position. Pick a Template (conventional, T-tail, V-tail, canard, flying wing, biplane, sailplane) to fill every field with an illustrative aircraft of that layout, then change what differs. The templates are not real types.
• Advanced mode adds the remaining options and a Geometry From choice: the parametric aircraft, the geometry grids, or an AVL file. After every calculation the grids are rewritten with the aircraft that was analysed. To refine a parametric aircraft or an imported file beyond what the fields allow, calculate once, set Geometry From to Geometry grids and edit the rows.
• The grids are plain rows of cells separated by commas, semicolons or tabs, so a block copied from a spreadsheet pastes directly. A header row and lines starting with # are skipped. Lengths are in the unit of the Grid Lengths In choice and are not converted when the unit system of the form is switched.

Grid	Cells of a row
Surfaces	name, mirror (yes / no), chordwise panels, spanwise panels of the entered half, chordwise spacing, spanwise spacing, section cl,max (- for none), cd0, k, cl0 of the strip polar cd = cd0 + k (cl − cl0)²
Sections	surface, xLE, y, z, chord, incidence (deg), airfoil (NACA four digits or flat), then three cells per control on that section: name, hinge x/c, sign of the mirrored side
Body stations	body, x, radius, z of the axis; on the first row optionally y of the axis and yes to mirror the body

Sections run from root to tip. A control surface exists between two neighbouring sections that both carry its name. The mirrored side moves by the sign times the deflection of the entered side: +1 for an elevator or a flap, −1 for an aileron. Writing the name as name*−1 reverses a control. Spacings are uniform, cosine (panels bunched at both ends), sine (bunched at the first section or the leading edge) and -sine (bunched at the last section: the tip of a half wing entered root to tip).

Axes and signs

• Geometry: x aft, y to the right (starboard), z up. The templates put the origin at the fuselage nose.
• Flight state: angle of attack α of the x axis to the flight path; sideslip β positive with the wind from the right. Rates are nondimensional, p b / 2V, q c / 2V and r b / 2V, about body axes x forward, y right, z down through the centre of gravity.
• Coefficients: stability axes. C_L and C_D are normal to and along the flight path in the plane of symmetry; C_Y to the right; C_l positive right wing down, C_m positive nose up, C_n positive nose right. Forces are divided by q S, the pitching moment by q S c, rolling and yawing moments by q S b.
• Controls: a positive deflection is trailing edge down on the entered (right) side. A positive elevator therefore lowers the nose, a positive aileron (right down, left up) rolls to the left, and a positive rudder (trailing edge to the right) turns the nose to the right.
• Reference quantities: unless entered, S, c (the mean aerodynamic chord), b and the moment reference point (the area-weighted quarter-chord point) come from the first surface. For a straight-tapered wing of span b, root chord c_r and taper ratio λ these are S = b c_r (1 + λ) / 2 and c = (2/3) c_r (1 + λ + λ²) / (1 + λ), at the spanwise station (b/6)(1 + 2λ) / (1 + λ).

The lattice

Each surface is divided into panels on its chord planes (the planes through the section leading edges with the chords along x). Camber, incidence, twist and control deflection do not move the panels; they turn the panel normals in the boundary condition, which is the thin-surface approximation of linear theory. A NACA four-digit number gives the camber line; its thickness digits are not used.

Every panel carries a horseshoe vortex: a bound segment on the panel quarter-chord line and two trailing legs that run downstream to infinity along the x axis. The wake is therefore flat and fixed to the aircraft, not aligned with the wind and not rolled up; this is the usual small-angle assumption and it makes the solution linear in the flight state. The flow-tangency condition is applied at the three-quarter-chord point of each panel, at the middle of the strip in the spacing variable (for cosine spacing this is what makes the solution converge quickly, as Hough shows in NASA SP-405).

The unknown circulations follow from one factorisation of the influence matrix. An aircraft that is symmetric about its centre plane is solved as a symmetric and an antisymmetric half problem; other geometries are solved whole. Limits: 600 panels in all for a symmetric aircraft, 400 otherwise. The solution is stored as the response to unit onset components (three velocity components, three rotation rates, each alone and with each control), so any flight state is an exact combination of them.

Forces, drag and derivatives

• Forces come from the Kutta-Joukowski law on the bound segments, F = ρ Γ V × l, with V the local velocity including the velocity induced by all the other vortices and the fuselage.
• Induced drag is given twice: from the forces on the bound vortices (near field) and from the wake far downstream (Trefftz plane). On an unswept planar wing they agree within half a percent. On swept surfaces the near-field value converges slowly with panel count and is low; the Trefftz-plane value is the one the tool uses for the total drag, the span efficiency e = C_L² / (π A C_Di) and the polar.
• Profile drag is the sum over the strips of the entered section polar at each strip's lift coefficient, plus the entered drag area of the bodies and any extra drag coefficient. It is an estimate that depends entirely on those entered values.
• Derivatives are the exact derivatives of the solution at the analysed state, not finite differences: the force is a quadratic form in the onset components and that form is differentiated. They are given per radian and per unit of nondimensional rate. The derivatives with respect to the rate of change of α and to speed are outside a steady lattice.
• Neutral point: the point about which the pitching moment does not change with α; the static margin is its distance behind the centre of gravity divided by c.
• Compressibility: the Prandtl-Glauert (Göthert) rule, sound to about Mach 0.7 and refused from Mach 0.9.
• Ground effect: an image of the lattice below a ground plane parallel to the x-y plane (valid at small α).

Trim, stall onset, polar and loads

The trimmed analyses solve the angle of attack and the pitch control by Newton's method so that the lift equals the weight times the load factor and the pitching moment about the centre of gravity is zero. A pull-up adds the pitch rate q = g (n − 1) / V; a coordinated level turn at bank angle φ has n = 1 / cos φ and the pitch and yaw rates of the turn rate g tan φ / V. An entered asymmetric yawing moment (an engine out) is balanced with aileron and rudder as well.

Stall onset is the angle of attack at which the first strip reaches the entered section c_l,max, with the lift of the aircraft at that angle. It marks where the linear solution stops being valid; the real maximum lift is somewhat higher, and whether the wing root, the wing tip, the tail or the canard gets there first is itself useful design information. The polar sweeps the angle of attack and adds induced, profile and body drag. The structural loads are the air loads on the wing integrated from the tip inward: shear, bending moment and torsion about the quarter-chord line, without inertia relief.

The fuselage model and its limits

A fuselage is a body of revolution described by stations (x, radius, height of the axis). It acts in two ways.

• On the lifting surfaces: sources along the axis (the displacement of the flow by the body's thickness) and doublets along the axis (the body in crossflow) add their velocities at the lattice. This is what gives the upwash on the wing beside the fuselage and the change in the flow at the tail.
• Its own forces: slender-body theory, with the apparent-mass factor k₂ − k₁ that Munk gives in NACA Report 184 as a function of fineness ratio. A closed body has no lift and no drag of its own in this theory but a destabilising moment, Munk's moment, equal to 2 q (k₂ − k₁) times the volume per radian in pitch and likewise in yaw. This is why adding the fuselage moves the neutral point forward and lowers C_nβ.
• Wing carry-over (an option): where a lifting surface passes through the body, the crossflow that loads the body is taken as blocked by that surface over its chord (Multhopp's rule). It is one-way: the lift carried across the fuselage is represented by the wing lattice running through the body.

Limits: bodies of circular section only (use the radius of the circle of equal area); no viscous crossflow lift or drag, so the fuselage contribution at large angles is not modelled; the body does not react to the lattice. For the default aircraft the fuselage pitching-moment slope is about two thirds of the value from Multhopp's strip method in the Longitudinal Static Stability tool. Treat the fuselage terms as correct in sign and order of magnitude. The results always show the totals, the derivatives and the neutral point with and without the fuselage so that its share can be judged.

The derivative set

The export box holds the "anowysis-derivative-set" (version 1): reference quantities, flight condition, mass and inertias, the static values and all derivatives in SI units and radians, with the moments about the reference point. Paste it into the Aircraft Stability Analysis tool to obtain the short-period, phugoid, Dutch-roll, roll and spiral modes, the flying-quality Levels and time histories. Positions in the set are in the geometry axes (x aft, z up). The inertias are entered about body axes and written turned to the stability axes of the analysed α. Derivatives a steady lattice cannot give are written as null, and that tool estimates them and says so.

What was verified

Case	Source	Printed	This solver
Rectangular wing, aspect ratio 2: CLα	NASA SP-405, p. 328	2.474	2.4743
Warren-12 swept planform: CLα; centre of pressure	NASA SP-405, pp. 330–331	2.74 to 2.75; 0.751	2.750; 0.7506
Cropped double delta at Mach 0.54: CLα, CLq, Cmq, Clp	NASA TN D-6142, sample cases 5 to 7	1.79879, 1.75972, −0.73161, −0.12986	within 0.1 to 1.5 %
Ground effect on induced drag	NACA TM 77, pp. 5 and 7	Prandtl's formula and example	within 1 %
Apparent-mass factors	NACA Report 184, Table I	table	reproduced

Checked against closed forms and an independent numerical implementation only, because no public document prints such a case: the two-dimensional limits of thin-airfoil theory (lift-curve slope 2π, flap effectiveness, camber), lifting-line comparisons, the Prandtl-Glauert rule, Munk's moment, the trim solution, loads, stall onset, the polar and hinge moments; twenty seeded configurations agree with the independent implementation to 2 × 10⁻¹², and the derivatives agree with finite differences of complete solutions to 6 × 10⁻⁸. The page itself is checked on every template, every analysis kind, both unit systems, the grids and the AVL import, and its derivative set is read back by the Aircraft Stability Analysis tool.

Limits

• Attached, inviscid, small-disturbance flow: results beyond about 10° of angle of attack or sideslip, or beyond the stall onset reported, are outside the method.
• Thin surfaces: no thickness effects, no leading-edge suction loss, no viscous decambering, so lift-curve slopes tend to be a little high and control effectiveness is above measured values, often noticeably.
• Flat wake along the x axis: no wake roll-up and no wake displacement with α, which matters for tails in the wake of a wing at high lift and for closely coupled canards.
• Hinge moments come from forces lumped on the bound vortices and are estimates.
• Not modelled: propeller slipstream and jet effects, leading-edge devices, external airfoil files, non-circular bodies, aeroelastic deformation.