3Daerodynamics

3D aerodynamic lifting surface solver suite featuring Linear and Coupled Non-Linear Vortex Lattice Methods (VLM and NL-VLM) with decambering and automated airfoil polars.

3Daerodynamics is an advanced 3D aerodynamic solver suite designed to calculate the lift, circulation distributions, and induced and profile drag of custom lifting surfaces using panel methods. It incorporates two principal solvers:

  1. Linear Vortex Lattice Method (VLM) for potential flow analysis.
  2. Coupled Non-Linear Vortex Lattice Method (NL-VLM) utilizing the decambering method linked to 2D viscous airfoil polar curves.

Toolchain & Geometry Parameterization

The suite provides a modular pipeline integrating geometric wing station parameterization, automatic airfoil coordinate fetching, meshing, and VTU unstructured grid exports:

  • Automatic Airfoil Lookup & Caching: Connects dynamically to the UIUC Airfoil Coordinates Database (AirfoilTools) to fetch coordinates and polar performance datasets (\(\text{Re} = 1{,}000{,}000\)), caching them locally for offline execution.
  • Camber & Polar Fitting: Airfoil mean camber lines \(z_c(x)\) are fitted with 4th-degree polynomials, while viscous lift \(C_l(\alpha)\) and profile drag \(C_{d,\text{p}}(\alpha)\) are modeled using cubic spline interpolation to capture stall and the drag bucket without spurious oscillations.
  • VTU Exports: Direct unstructured XML exports compatible with ParaView for 3D field rendering of circulation \(\Gamma\), induced velocity \(\mathbf{V}_{\text{ind}}\), and surface forces.

1. Linear Vortex Lattice Method (VLM)

Discretizing the thin lifting surface into a grid of \(N\) quadrilateral panels, each panel \(n\) contains a horseshoe vortex with a bound segment along the 1/4-chord line and trailing semi-infinite legs.

Biot-Savart Formulation

The induced velocity \(\mathbf{V}_{\text{ind}}\) at control point \(m\) (at the 3/4-chord midpoint) by straight vortex segment \(\mathbf{r}_0 = \mathbf{r}_2 - \mathbf{r}_1\) of strength \(\Gamma\) is evaluated via Biot-Savart:

\[\mathbf{V}_{\text{ind}} = \frac{\Gamma}{4\pi} \frac{\mathbf{r}_1 \times \mathbf{r}_2}{\lVert \mathbf{r}_1 \times \mathbf{r}_2 \rVert^2} \left[ \mathbf{r}_0 \cdot \left( \frac{\mathbf{r}_1}{\lVert \mathbf{r}_1 \rVert} - \frac{\mathbf{r}_2}{\lVert \mathbf{r}_2 \rVert} \right) \right]\]

Enforcing flow tangency at each panel control point (\(\mathbf{V}_m \cdot \mathbf{n}_m = 0\)):

\[\sum_{n=1}^{N} A_{mn} \Gamma_n = -\mathbf{V}_{\infty} \cdot \mathbf{n}_m\]

where \(A_{mn} = \mathbf{C}_{mn} \cdot \mathbf{n}_m\) is the aerodynamic influence coefficient matrix. Local panel forces are calculated via the Kutta-Joukowski theorem on the bound vortex:

\[\mathbf{F}_m = \rho_{\infty} \Gamma_m \left( \mathbf{V}_{\text{ind}, m} \times \mathbf{r}_{0, m} \right)\]

2. Non-Linear VLM via Iterative Decambering

Potential flow inherently neglects viscosity, flow separation, and stall. The NL-VLM couples the 3D potential solver with experimental/viscous 2D sectional polars through the Decambering Method:

Iterative Matching

  1. Effective Angle of Attack: \(\alpha_{\text{eff}}(y) = \alpha_{\infty} + \theta(y) - \alpha_{\text{i}}(y)\) where downwash angle \(\alpha_{\text{i}}(y)\) is computed from the spanwise circulation gradient \(\frac{d\Gamma}{dy}\).

  2. Decambering Update: Comparing potential lift \(C_{L, \text{VLM}}(y) = \frac{2\Gamma_{\text{sec}}(y)}{V_{\infty} c(y)}\) to target viscous polar lift \(C_{l, \text{target}}(\alpha_{\text{eff}})\):

    \[\delta_{\text{decamber}}^{(k+1)} = \delta_{\text{decamber}}^{(k)} + \omega \frac{C_{L, \text{VLM}} - C_{l, \text{target}}(\alpha_{\text{eff}})}{2\pi}\]

    where \(\omega \in [0.02, 0.05]\) is a relaxation factor modifying panel normal boundary conditions until convergence.

  3. Integrated Wing Drag: Total wing drag combines 3D induced drag with integrated sectional profile drag:

    \[C_D = C_{D,\text{i}} + \frac{1}{S} \sum_{i=1}^{N_{\text{span}}} C_{d,\text{p}, i} c_i \Delta y_i\]