SA1 — Aircraft Wing Analysis & Aerofoil Design

Build a fast numerical method to compare and design efficient 2D aerofoil sections at two very different operating conditions, optimising lift-to-drag ratio (L/D):

The design question: how should the aerofoil pressure loading change when boundary-layer behaviour changes with Reynolds number?

The solver was built up rather than started from a black box: the point-vortex streamfunction, then a linearly varying vortex-sheet panel (verified against a discretised set of point vortices), then generalised to an arbitrary panel — producing the reusable panelinf() routine.

The surface-streamline condition plus trailing-edge conditions close the system around the geometry.

Before aerofoils, the panel method was tested on cylinder flow — surface velocity distribution, streamlines and circulation — a case with known physical behaviour, so the numerics were checked before moving to harder geometries.

The analysis pipeline reads the surface, builds a high-resolution spline, normalises to unit chord, repanels, solves the potential-flow system, and extracts circulation and surface pressure. It locates the stagnation point and splits the upper and lower surfaces before the viscous solver.

Cp = 1 − (ue / U)²

The boundary-layer model was built up progressively:

Thwaites' integral method predicts laminar boundary-layer development. The zero-pressure-gradient case was checked against the Blasius solution — implementation, then reference comparison, then confidence.

The laminar solver detects whether the layer naturally transitions (from a momentum-thickness / shape-factor criterion) or laminar-separates (from the Thwaites pressure-gradient parameter). Reynolds number and pressure gradient move that location — and the final designs depend strongly on where the upper-surface laminar layer breaks down.

Past transition or laminar separation, the solver switches to a turbulent integral model, evolving momentum and energy thickness through coupled ODEs integrated with MATLAB ode45, and detects turbulent reattachment and separation from the energy shape factor.

Natural transition, laminar separation, turbulent reattachment and turbulent separation are tracked separately on the upper and lower surfaces.

One-way coupled: the panel solver produces the edge velocity / pressure field and the boundary layer responds to it. Displacement thickness does not feed back into the potential-flow solution.

Lift comes from the inviscid circulation; profile drag is estimated from the boundary-layer state near the trailing edge using the Squire–Young wake relation — so momentum-thickness growth drives the whole design process.

Before designing anything, the combined model was validated against reference data for NACA0012 and NACA4412 at ~Re = 3×10⁶ — lift curve, drag polar, Cd vs incidence and L/D vs incidence.

The NACA0012 test produced numerical symmetry residuals of ~10⁻¹⁴, confirming consistent panel ordering, sign conventions, geometry handling and force extraction.

But the computed lift-curve slope was systematically too high — NACA0012 +9.5%, NACA4412 +14.4% — so not every computed operating point was treated as trustworthy.

The validated incidence range used for quantitative design conclusions. Outside it, the inviscid model keeps producing strong attached-flow circulation even where the real flow turns nonlinear — so stall angle, maximum Cl and post-stall behaviour were deliberately not claimed.

Comparing NACA0012 / 2412 / 4412 / 6412 (all 12% thickness, increasing camber): NACA6412 was chosen as the baseline for strong lift and good L/D at modest incidence while staying inside the validated region.

Reducing thickness (12% → 9%, same camber) to cut the pressure-recovery / viscous penalty. It was numerically better in isolation:

Not selected. At Re = 5×10⁵, α = 4°, the thinner section developed upper-surface laminar separation near the leading edge (x/c ≈ 0.022 vs ≈ 0.375 for NACA6412) — a narrow, fragile efficient range. Lower geometric thickness did not mean lower useful drag once the boundary layer was considered.

Moving maximum camber from 40% to 30% chord (same 6% camber, 12% thickness) shifted loading forward. The same change had opposite effects at the two Reynolds numbers:

Low-Re: stronger forward suction forced an earlier adverse recovery (harmful). High-Re: the stronger loading paid off.

Developed from NACA6412 with the project's interactive surface generator. The winning strategy was a smoother, more aft-loaded upper-surface suction — not simply more camber or less thickness — spreading the loading and delaying the strong adverse gradient the laminar layer sees.

The baseline upper-surface laminar separation at x/c ≈ 0.375 was delayed to x/c ≈ 0.500, then reattached at x/c ≈ 0.541. The delayed disturbance kept upper-surface momentum thickness lower across much of the chord and reduced wake momentum loss.

A different problem: transition is early, so preserving a long laminar run is no longer the goal. The section instead carries stronger sustained suction over the front and middle of the upper surface while keeping turbulent boundary-layer growth controlled.

NACA6312 produced a slightly higher numerical maximum (170.9) — but that optimum carried a turbulent-separation flag. The final design was chosen because its optimum stayed inside the validated incidence range and did not rely on the less trustworthy separated-flow regime. The largest number was not selected blindly.

The preferred pressure distribution changed because the boundary-layer physics changed — not because of a single universal optimum.

For fixed geometry and incidence, the inviscid Cp distribution is Reynolds-number independent. Re enters through the viscous calculation — affecting transition, laminar separation, reattachment, turbulent development, momentum thickness, Cd and L/D — rather than changing the potential-flow pressure field directly.

Viscous and inviscid solvers are one-way coupled; transition uses an empirical criterion; surface roughness, freestream turbulence and manufacturing effects are not explicitly modelled.