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Aerodynamics Theory

Vortex Lattice Method (VLM)

The Vortex Lattice Method (VLM) is a low-fidelity numerical method used to predict aerodynamic loads on lifting surfaces. It represents the wing as a collection of horseshoe vortex elements.

Paneling

To resolve high-gradient regions at leading and trailing edges, the library uses cosine spacing for chordwise and spanwise distributions:

\[x_i = \frac{1}{2} [ 1 - \cos( \frac{i \cdot \pi}{N} ) ]\]

for \(i = 0, \dots, N\). This distribution clusters panels near the boundaries where pressure gradients are steepest (Katz & Plotkin, 2001).

Horseshoe Vortex Convention

Each panel \(j\) has a horseshoe vortex consisting of:

  • Bound vortex segment: \(A_j \rightarrow B_j\) (inboard to outboard at quarter-chord)
  • Right trailing leg: \(B_j \rightarrow +\infty\) in freestream direction, strength \(+\gamma\)
  • Left trailing leg: \(A_j \rightarrow +\infty\) in freestream direction, strength \(-\gamma\)

Biot-Savart Law

The velocity induced at a point \(P\) by a vortex segment is given by the Biot-Savart law. For a finite segment \(A \rightarrow B\):

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

where \(\mathbf{r}_0 = \mathbf{B} - \mathbf{A}\), \(\mathbf{r}_1 = \mathbf{P} - \mathbf{A}\), and \(\mathbf{r}_2 = \mathbf{P} - \mathbf{B}\).

Rankine Core Regularisation

To prevent singularities when a control point lies on or near a vortex axis, a Rankine core regularisation is used. The induced velocity denominator is bounded below:

\[ \text{denom} = \max(|\mathbf{r}_1 \times \mathbf{r}_2|^2, (\epsilon \cdot |\mathbf{r}_0|)^2) \]

where \(\epsilon\) is a small fraction (typically \(10^{-6}\)).

Aerodynamic Influence Coefficient (AIC) Matrix

The AIC matrix relates the circulation \(\gamma_j\) of each panel to the normal velocity induced at control points \(CP_i\):

\[ \mathbf{AIC} \cdot \mathbf{\gamma} = \mathbf{RHS} \]

The no-penetration boundary condition at each control point \(i\) states:

\[ (\mathbf{V}_{\infty} + \mathbf{v}_{\text{induced}}) \cdot \hat{\mathbf{n}}_i = 0 \]
\[ \mathbf{RHS}_i = -(\mathbf{V}_{\infty} \cdot \hat{\mathbf{n}}_i) \]

Static Aeroelasticity

The aeroelastic discipline couples the VLM aerodynamic solver with an Euler-Bernoulli structural beam model.

Coupling Algorithm

  1. Rebuild WingGeometry with current vertical deflection \(z\) and elastic twist.
  2. Run VLMDiscipline to obtain spanwise load and drag.
  3. Run StructuralDiscipline to obtain spanwise deflection and twist.
  4. Check convergence; if not converged (max change in deflection \(> 0.01\) m), repeat.

The elastic twist (aeroelastic wash-out) captures the dominant load-relief mechanism on swept wings.

Force Integration

Near-Field Forces (Kutta-Joukowski)

Total forces are obtained by integrating the Kutta-Joukowski force on each bound vortex segment:

\[ \mathbf{F}_j = \rho \Gamma_j (\mathbf{V}_{\text{local},j} \times \mathbf{dl}_j) \]

Where \(\mathbf{V}_{\text{local},j}\) is the local velocity vector at the midpoint of the bound segment, which includes the freestream velocity and the induced downwash from the entire vortex system (Katz & Plotkin, 2001).

Trefftz-Plane Induced Drag

Induced drag is also computed in the Trefftz plane (far downstream) for better accuracy by evaluating the momentum deficit in the far-field:

\[ D_i = \frac{1}{2} \sum_{i=1}^{N} \Gamma_i \cdot w_{i, \infty} \cdot \Delta y_i \]

Note that the far-field normal wash \(w_{i, \infty}\) is twice the bound-vortex downwash at the wing (Drela, 2014). For three-dimensional systems with non-planar wakes, the coefficient form is used:

\[ C_{Di} = -\frac{1}{V_{\infty}^2 S_{\text{ref}}} \sum \Gamma_j (w_{tp,j} \Delta y_j - v_{tp,j} \Delta z_j) \]

Profile Drag

Zero-lift profile drag is estimated using turbulent flat-plate strip theory with a thickness form factor:

\[ C_{f} = \frac{0.074}{Re_c^{0.2}} \]
\[ FF = 1 + 2 \frac{t}{c} \]
\[ c_{d0} = 2 C_f FF \]

References

  • Katz, J. and Plotkin, A. "Low-Speed Aerodynamics", 2nd ed., Cambridge University Press, 2001.
  • Drela, M. "Flight Vehicle Aerodynamics", MIT Press, 2014.
  • Hodges, D.H. and Pierce, G.A., "Introduction to Structural Dynamics and Aeroelasticity", 2nd ed., Cambridge, 2011.